Helsinki

University

of

Technology

Department

of

Computer

Science

Digital

Systems

Laboratory

Otaniemi

Otakaari

SF

ESPOO

FINLAND

HELSINKI

UNIVERSITY

OF

TECHNOLOGY

DIGITAL

SYSTEMS

LABORATORY

Series

A

Research

Reports

ISSN

No.

September

ISBN

CONSTRUCTION

METHODS

FOR

COVERING

CODES

Patric

R.

J.

sterg?rd

Digital

Systems

Laboratory

Department

of

Computer

Science

Helsinki

University

of

Technology

SF

Espoo

Finland

HELSINKI

UNIVERSITY

OF

TECHNOLOGY

DIGITAL

SYSTEMS

LABORATORY

Series

A

Research

Reports

ISSN

No.

September

ISBN

Construction

Methods

for

Covering

Codes

Patric

R.

J.

sterg?rd

Abstract

A

covering

code

in

a

Hamming

space

is

a

set

of

codewords

with

the

property

that

any

word

in

the

space

is

within

a

specified

Hamming

distance

the

covering

radius

from

at

least

one

codeword

In

this

thesis

efficient

construction

methods

for

such

codes

are

considered

The

constructions

work

with

some

exceptions

for

codes

over

alphabets

consisting

of

any

number

of

symbols

Codes

over

mixed

alphabets

are

also

discussed

Most

of

the

methods

are

developed

in

order

to

determine

values

of

Kq

-LRB-

n

R

-RRB-

the

minimumnumber

of

codewords

in

a

q-ary

code

of

length

n

and

covering

radius

R.

Codes

obtained

by

the

constructions

prove

upper

bounds

on

this

function

In

many

of

the

constructions

simulated

annealing

a

probabilistic

optimization

method

has

turned

out

to

perform

very

well

Simulated

annealing

can

not

be

used

to

prove

optimality

of

codes

found

in

that

case

the

problem

is

viewed

and

solved

as

a

set

covering

problem

For

larger

codes

a

direct

approach

is

not

generally

feasible

it

is

shown

how

good

such

codes

can

be

found

by

combining

existing

codes

or

by

imposing

some

structure

on

the

codes

A

matrix

method

that

is

presented

follows

the

latter

principle

a

code

constructed

by

this

method

consists

of

cosets

of

a

linear

code

To

be

able

to

combine

existing

codes

efficiently

it

is

necessary

that

the

codes

involved

can

be

partitioned

into

subcodes

with

certain

properties

subnorm

and

norm

are

concepts

that

are

introduced

to

describe

partitions

of

codes

Finally

some

families

of

combinatorial

methods

are

presented

Dissertation

for

the

degree

of

Doctor

of

Technology

to

be

presented

with

due

permission

for

public

examination

and

debate

in

Auditorium

L

at

Helsinki

University

of

Technology

-LRB-

Espoo

Finland

-RRB-

on

the

of

September

at

o'clock

noon

Printing

TKK

Monistamo

Otaniemi

Helsinki

University

of

Technology

Phone

-LCB-

Department

of

Computer

Science

Digital

Systems

Laboratory

Telex

htkk

sf

Otaniemi

Otakaari

Telefax

-LCB-

-LCB-

SF

-LCB-

ESPOO

FINLAND

E-mail

lab@hutds.hut.fi

i

Preface

The

work

covered

in

this

thesis

has

been

carried

out

at

the

Digital

Systems

Laboratory

of

Helsinki

University

of

Technology

during

I

am

grateful

to

the

head

of

the

laboratory

Prof.

Leo

Ojala

for

providing

excellent

working

conditions

and

for

supporting

me

during

this

period

My

thanks

also

go

to

all

colleagues?former

and

present?for

the

inspiring

and

intellectual

atmosphere

in

our

laboratory

I

especially

wish

to

thank

Dr.

Nisse

Husberg

for

his

supervising

help

when

I

wanted

to

take

part

in

the

research

activities

of

the

laboratory

back

in

I

must

express

special

gratitude

to

Dr.

Iiro

Honkala

for

his

help

in

my

work

I

have

been

very

fortune

to

get

his

valuable

comments

on

many

of

my

preliminary

results

a

thick

folder

consisting

of

his

letters

will

always

remind

me

of

our

numerous

fruitful

discussions

I

very

much

appreciate

the

discussions

with

Heikki

H?m?l?inen

and

Markku

Kaikkonen

on

football

pool

systems

and

other

coding

theory

issues

It

was

a

true

pleasure

to

co-author

one

of

the

papers

in

this

thesis

with

Heikki

I

thank

Elizabeth

Heap-Talvela

for

checking

the

English

language

of

the

thesis

I

also

thank

the

Foundation

of

Technology

the

Engineering

Society

in

Finland

TFiF

the

Oscar

AEund

Foundation

the

Ella

and

Georg

Ehrnrooth

Foundation

the

Helsingin

Sanomat

Centenary

Foundation

and

the

Academy

of

Finland

for

their

onancial

support

during

this

work

Finally

I

am

very

grateful

to

my

family

and

my

friends

who

have

all

supported

me

throughout

this

work

Espoo

August

Patric

sterg?rd

ii

Contents

Preface

i

List

of

Publications

iii

Introduction

Covering

Codes

Nonmixed

Covering

Codes

Mixed

Covering

Codes

Computer

Search

for

Covering

Codes

A

Set

Covering

Approach

A

Probabilistic

Approach

Partitioning

Constructions

Normal

Seminormal

and

Subnormal

Codes

Combining

Codes

A

Matrix

Method

Other

Combinatorial

Methods

Some

Direct

Constructions

Piecewise

Constant

Codes

Lengthening

Constructions

Conclusions

References

Errata

iii

List

of

Publications

This

thesis

consists

of

a

summary

and

the

following

publications

-LSB-

-RSB-

P.

R.

J.

sterg?rd

A

new

binary

code

of

length

and

covering

radius

IEEE

Transactions

on

Information

Theory

-LRB-

-RRB-

-LSB-

-RSB-

P.

R.

J.

sterg?rd

Upper

bounds

for

q-ary

covering

codes

IEEE

Transactions

on

Information

Theory

-LRB-

-RRB-

and

-LRB-

-RRB-

-LSB-

-RSB-

P.

R.

J.

sterg?rd

Further

results

on

-LRB-

k

t

-RRB-

subnormal

covering

codes

IEEE

Transactions

on

Information

Theory

-LRB-

-RRB-

-LSB-

-RSB-

P.

R.

J.

sterg?rd

Construction

methods

for

mixed

covering

codes

in

M.

Gyllenberg

and

L.

E.

Persson

-LRB-

eds

-RRB-

Proceedings

of

the

Nordic

Congress

of

Mathematicians

Marcel

Dekker

New

York

to

appear

-LSB-

-RSB-

P.

R.

J.

sterg?rd

New

upper

bounds

for

the

football

pool

problem

for

and

matches

Journal

of

Combinatorial

Theory

Series

A

to

appear

-LSB-

-RSB-

P.

R.

J.

sterg?rd

and

H.

O.

H?m?l?inen

New

upper

bounds

for

binary/ternary

mixed

covering

codes

Research

Report

A

Digital

Systems

Laboratory

Helsinki

University

of

Technology

Papers

-LSB-

-RSB-

to

-LSB-

-RSB-

were

written

without

co-authors

The

mathematical

theory

in

-LSB-

-RSB-

was

developed

by

the

author

of

the

thesis

who

also

wrote

the

paper

The

ideas

behind

the

constructions

and

the

new

codes

in

the

same

paper

are

due

to

both

authors

Introduction

In

Taussky

and

Todd

-LSB-

-RSB-

considered

the

following

group-theoretic

problem

Let

G

be

an

Abelian

group

with

n

independent

generators

gn

each

of

order

q

A

subset

H

of

G

is

said

to

cover

G

if

for

every

g

G

there

is

an

h

H

such

that

g

hgi

for

some

i

n

q

The

problem

is

now

for

given

n

and

q

to

ond

the

minimum

cardinality

of

a

covering

set

H

oe

-LRB-

n

q

-RRB-

If

it

is

further

required

that

H

be

a

subgroup

of

G

the

minimum

cardinality

is

denoted

by

oe

-LRB-

n

q

-RRB-

In

this

group-theoretic

context

the

problem

has

been

discussed

in

a

variety

of

papers

for

example

-LSB-

-RSB-

Actually

long

before

the

paper

-LSB-

-RSB-

by

Taussky

and

Todd

appeared

there

was

a

big

interest

in

this

problem

among

football

pool

enthusiasts

in

the

Nordic

countries

In

the

football

pool

game

you

have

to

guess

the

outcome

of?usually

or

football

matches

For

each

match

there

are

three

alternatives

you

mark

a

if

you

think

that

the

home

team

will

win

an

x

if

you

think

that

the

teams

will

draw

or

a

if

you

think

that

the

away

team

will

win

If

your

guess

is

correct

you

win

the

orst

prize

if

the

outcome

of

exactly

one

match

dioeers

from

your

guess

you

win

the

second

prize

etc.

Usually

there

are

or

prize

classes

Now

the

following

combinatorial

question

follows

very

naturally

If

there

are

n

matches

on

a

football

pool

coupon

what

is

the

minimum

number

of

guesses

-LRB-

by

a

guess

is

still

meant

an

estimate

of

the

outcome

of

all

matches

-RRB-

such

that

for

any

outcome

of

the

matches

you

will

win

at

least

the

jth

prize

A

collection

of

guesses

is

usually

called

a

football

pool

system

For

further

discussion

of

the

early

history

of

such

systems

see

-LSB-

-RSB-

The

aforementioned

problem

of

constructing

football

pool

systems

is

more

general

than

the

original

problem

proposed

by

Taussky

and

Todd

Namely

a

set

H

G

must

be

found

which

has

the

property

that

for

every

g

G

there

is

an

h

H

such

that

g

hg

g

g

j

aj

for

some

aj

n

i

q

A

football

pool

system

is

now

obtained

from

a

covering

set

H

by

encoding

all

elements

according

to

for

example

the

following

scheme

x

and

As

a

matter

of

fact

the

source

of

inspiration

for

research

in

this

area

was

from

the

very

beginning

the

problem

of

constructing

football

pool

systems

Taussky

and

Todd

-LSB-

-RSB-

refer

to

this

problem

as

a

certain

industrial

problem

however

in

a

paper

by

Mattioli

-LSB-

-RSB-

published

in

the

connection

is

explicitly

stated

Given

a

set

G

and

the

covering

properties

that

have

to

be

fulolled

the

problem

of

onding

the

minimum

cardinality

of

a

covering

set

H

is

far

from

trivial

Exact

values

have

been

obtained

only

in

some

special

cases

mostly

for

small

values

of

the

parameters

In

the

general

case

eoeorts

have

been

concentrated

on

obtaining

good

upper

and

lower

bounds

on

the

minimum

cardinality

Good

lower

bounds

can

be

obtained

by

a

variety

of

analytic

methods

-LSB-

-RSB-

upper

bounds

on

the

other

hand

are

found

by

explicitly

constructing

a

covering

set

In

this

work

new

construction

methods

are

developed

thereby

improving

many

upper

bounds

A

survey

of

construction

methods

for

covering

codes

was

given

by

Cohen

et

al.

in

-LSB-

-RSB-

some

later

results

were

surveyed

by

Van

Lint

in

-LSB-

-RSB-

In

this

thesis

these

surveys

are

not

only

brought

up-to-date

by

including

the

results

in

-LSB-

-RSB-

to

-LSB-

-RSB-

and

others

new

results

but

new

light

is

also

shed

on

some

old

methods

In

Section

an

exact

mathematical

formulation

-LRB-

in

a

combinatorial

context

-RRB-

of

the

problem

is

given

The

construction

methods

are

presented

in

Sections

to

These

are

divided

into

the

following

groups

computer

search

for

covering

codes

is

discussed

in

Section

partitioning

constructions

are

given

in

Section

a

matrix

method

is

presented

in

Section

and

onally

some

other

combinatorial

methods

are

listed

in

Section

The

thesis

is

concluded

in

Section

Covering

Codes

In

the

earliest

papers

the

covering

problem

presented

in

the

Introduction

was

discussed

using

the

aforementioned

group-theoretic

notations

However

as

pointed

out

the

problem

can

also

be

considered

from

a

purely

combinatorial

point-of-view

In

the

's

discussions

of

the

covering

problem

in

a

combinatorial

context

became

more

and

more

common

-LRB-

-LSB-

-RSB-

etc.

-RRB-

with

the

years

this

has

become

the

predominant

approach

Consequently

this

approach

and

its

established

notations

are

used

throughout

this

thesis

There

should

however

be

no

problem

in

seeing

the

one-to-one

correspondence

between

the

two

approaches

In

this

section

codes

over

spaces

where

all

coordinates

are

of

the

same

arity

are

orst

considered

Subsequently

codes

over

mixed

alphabets

mixed

covering

codes

are

discussed

For

a

thorough

introduction

to

coding

theory

see

-LSB-

-RSB-

Nonmixed

Covering

Codes

Let

Fnq

be

an

n-dimensional

space

over

Fq

an

alphabet

consisting

of

q

elements

If

q

is

a

prime

power

Fq

is

the

onite

oeld

of

q

elements

-LRB-

also

called

the

Galois

oeld

of

q

elements

GF

-LRB-

q

-RRB-

-RRB-

otherwise

Fq

is

the

set

of

integers

modulo

q

In

any

case

the

set

Fq

q

is

used

to

denote

the

elements

of

the

alphabet

A

word

in

this

space

is

an

n-tuple

x

-LRB-

xn

-RRB-

where

xi

Fq

such

a

word

is

often

written

in

a

shorter

form

xn

A

code

C

is

now

a

nonempty

subset

of

Fnq

OE

C

Fnq

The

elements

in

C

are

codewords

and

the

length

of

the

code

is

n.

Actually

in

some

cases

it

is

necessary

to

deone

codes

as

multisets

that

is

to

allow

multiple

occurrences

of

codewords

in

the

code

however

in

this

thesis

such

codes

are

not

considered

The

Hamming

distance

d

-LRB-

x

y

-RRB-

between

two

words

x

y

Fnq

is

the

number

of

coordinates

in

which

they

dioeer

The

weight

of

a

word

x

Fnq

is

wt

-LRB-

x

-RRB-

d

-LRB-

x

-RRB-

that

is

the

number

of

nonzero

coordinates

-LRB-

-LRB-

-RRB-

-RRB-

The

Hamming

distance

function

is

a

metric

since

for

all

x

y

z

Fnq

d

-LRB-

x

y

-RRB-

d

-LRB-

x

y

-RRB-

ioe

x

y

d

-LRB-

x

y

-RRB-

d

-LRB-

y

x

-RRB-

and

d

-LRB-

x

z

-RRB-

d

-LRB-

x

y

-RRB-

d

-LRB-

y

z

-RRB-

A

code

C

Fnq

is

said

to

cover

Fnq

with

covering

radius

R

if

every

word

in

Fnq

is

within

Hamming

distance

R

from

some

codeword

in

C

and

if

this

is

not

fulolled

for

R

Such

a

code

is

called

a

-LRB-

q

n

M

-RRB-

R

code

where

M

jCj

is

the

cardinality

of

the

code

If

all

words

in

Fnq

have

at

least

codewords

within

Hamming

distance

R

the

code

is

said

to

be

a

fold

multiple

covering

Multiple

coverings

are

discussed

for

example

in

-LSB-

-RSB-

Furthermore

the

notation

Kq

-LRB-

n

R

-RRB-

minfM

j

there

is

a

-LRB-

q

n

M

-RRB-

R

codeg

is

used

The

index

q

is

usually

omitted

in

the

binary

case

Other

notations

that

have

been

used

in

the

literature

include

oe

-LRB-

n

k

-RRB-

Kk

-LRB-

n

-RRB-

oen

-LRB-

n

-RRB-

-LRB-

for

example

in

-LSB-

-RSB-

see

also

the

Introduction

-RRB-

and

-LRB-

n

k

s

-RRB-

Kk

-LRB-

n

n

s

-RRB-

-LSB-

-RSB-

The

determination

of

-LRB-

n

-RRB-

is

usually

referred

to

as

the

football

pool

problem

see

-LSB-

-RSB-

and

-LSB-

-RSB-

A

-LRB-

q

n

M

-RRB-

R

code

with

M

Kq

-LRB-

n

R

-RRB-

codewords

is

called

optimal

In

the

earliest

papers

only

codes

with

covering

radius

were

treated

In

the

early

's

the

following

exact

values

of

Kq

-LRB-

n

-RRB-

were

known

Kq

-LRB-

-RRB-

q

and

Kq

-LRB-

-RRB-

d

-LRB-

-RRB-

-LSB-

-RSB-

Kq

-LRB-

-LRB-

qk

-RRB-

-LRB-

q

-RRB-

-RRB-

q

-LRB-

qk

-RRB-

-LRB-

q

-RRB-

k

if

q

is

a

prime

power

and

k

-LSB-

-RSB-

see

also

-LSB-

-RSB-

-LRB-

the

Hamming

codes

-RRB-

Kpq

-LRB-

q

-RRB-

pqqq

if

q

is

a

prime

power

-LSB-

-RSB-

see

also

-LSB-

-RSB-

-LRB-

-RRB-

and

-LRB-

-RRB-

-LSB-

-RSB-

-LRB-

-RRB-

-LSB-

-RSB-

-LRB-

-RRB-

-LSB-

-RSB-

-LRB-

-RRB-

-LSB-

-RSB-

and

-LRB-

-RRB-

-LSB-

-RSB-

The

fact

that

the

only

result

that

has

been

added

to

this

list

during

the

last

two

decades

is

-LRB-

-RRB-

k

-LSB-

-RSB-

clearly

indicates

the

complexity

of

calculating

exact

values

of

Kq

-LRB-

n

R

-RRB-

Having

determined

the

value

of

Kq

-LRB-

n

R

-RRB-

for

given

parameters

it

is

also

interesting

to

know

the

number

of

nonequivalent

optimal

covering

codes

with

these

parameters

Two

codes

C

and

C

are

equivalent

if

C

can

be

obtained

from

C

by

a

permutation

of

the

coordinate

positions

and

a

permutation

of

the

elements

of

Fq

in

each

coordinate

position

Stanton

and

KalbAEeisch

-LSB-

-RSB-

enumerated

all

binary

optimal

covering

codes

up

to

equivalence

for

R

and

n

the

fact

that

there

is

a

unique

-LRB-

-RRB-

covering

code?the

binary

Hamming

code

of

length

was

proved

in

an

early

paper

by

Zaremba

-LSB-

-RSB-

KalbAEeisch

and

Stanton

-LSB-

-RSB-

proved

that

the

ternary

Hamming

code

of

length

is

a

unique

-LRB-

-RRB-

covering

code

Recently

Kolev

-LSB-

-RSB-

constructed

all

nonequivalent

-LRB-

-RRB-

covering

codes

Finally

it

is

known

-LSB-

-RSB-

that

the

number

of

nonequivalent

-LRB-

n

k

-RRB-

codes

is

at

least

for

some

constant

c

By

considering

the

total

number

of

words

in

a

space

and

the

number

of

words

covered

by

a

codeword

in

the

space

the

following

well-known

sphere

covering

bound

is

obtained

Theorem

Kq

-LRB-

n

R

-RRB-

qn

PRi

-LRB-

q

-RRB-

i

n

i

The

number

of

codewords

in

a

code

divided

by

the

bound

in

Theorem

is

called

the

density

-LRB-

-RRB-

of

the

code

Apparently

If

q

is

a

prime

power

then

the

density

of

an

optimal

code

with

covering

radius

approaches

as

n

goes

to

inonity

-LSB-

-RSB-

see

also

-LSB-

-RSB-

A

covering

code

with

density

is

called

perfect

For

any

word

in

the

space

exactly

one

codeword

of

a

perfect

code

is

within

Hamming

distance

R

from

this

word

In

the

early

's

Tiet?v?inen

and

Van

Lint

showed

that

if

q

is

a

prime

power

then

the

only

nontrivial

perfect

codes

are

codes

with

the

same

parameters

as

the

Hamming

and

Golay

codes

see

-LSB-

-RSB-

for

a

survey

of

these

results

In

the

binary

case

an

n-dimensional

space

can

be

displayed

as

a

hypercube

of

dimension

n

-LRB-

an

n-cube

-RRB-

There

is

now

a

one-to-one

mapping

between

the

nodes

in

the

n-cube

and

the

words

in

the

space

so

that

two

nodes

are

adjacent

exactly

when

the

corresponding

codewords

dioeer

in

one

coordinate

-LRB-

are

at

Hamming

distance

from

each

other

-RRB-

In

Figure

this

is

illustrated

for

n

The

black

nodes

of

the

graphs

indicate

the

codewords

of

a

-LRB-

-RRB-

code

Thus

K

-LRB-

-RRB-

since

by

Theorem

K

-LRB-

-RRB-

Figure

Hypercube

of

dimension

If

a

code

is

a

linear

subset

of

the

space

it

is

called

a

linear

code

A

linear

code

that

has

qk

-LRB-

q

is

a

prime

power

-RRB-

codewords

and

covers

Fnq

with

covering

radius

R

is

called

an

-LSB-

n

k

-RSB-

R

code

In

this

thesis

it

is

not

required

that

the

codes

be

linear

Upper

and

lower

bounds

on

binary

linear

covering

codes

have

been

considered

in

-LSB-

-RSB-

Mixed

Covering

Codes

Consider

a

space

H

Fnm

qm

with

m

ni

Pmi

ni

and

qi

If

m

and

there

are

indices

i

j

m

such

that

qi

qj

the

space

is

a

proper

mixed

space

Similarly

to

the

nonmixed

case

a

mixed

code

C

H

is

said

to

cover

H

with

covering

radius

R

if

every

word

in

H

is

within

Hamming

distance

R

from

some

codeword

in

C

and

if

this

is

not

fulolled

for

R

such

a

code

is

called

a

-LRB-

qm

nm

M

-RRB-

R

code

Furthermore

qm

-LRB-

nm

R

-RRB-

minfM

j

there

is

a

-LRB-

qm

nm

M

-RRB-

R

codeg

Results

for

nonmixed

codes

can

in

many

cases

directly

be

generalized

to

the

mixed

case

for

example

the

sphere

covering

bound

is

as

follows

Theorem

qn

-LRB-

R

-RRB-

-LRB-

Qni

qi

-RRB-

V

-LRB-

R

-RRB-

where

V

-LRB-

R

-RRB-

PRj

P

ij

n

-LRB-

-RRB-

-LRB-

qij

-RRB-

A

lot

of

research

has

been

done

on

mixed

covering

codes

however

for

a

long

period

of

time

research

only

considered

perfect

mixed

codes

Perfect

mixed

codes

have

been

discussed

for

example

in

-LSB-

-RSB-

During

the

last

decade

there

has

been

an

increasing

interest

in

mixed

covering

codes

especially

in

binary/ternary

codes

-LSB-

-RSB-

Binary/ternary

covering

codes

are

useful

as

football

pool

systems

if

one

of

the

three

possible

outcomes

for

some

matches

is

excluded

by

the

gambler

For

this

case

the

notation

K

-LRB-

t

b

R

-RRB-

-LRB-

R

-RRB-

is

used

in

-LSB-

-RSB-

The

construction

methods

to

be

presented

in

Sections

to

are

mainly

given

for

nonmixed

covering

codes

This

is

done

to

keep

the

presentation

plain

In

cases

where

the

generalization

of

a

method

to

the

mixed

case

is

not

clear

or

is

not

even

possible

this

is

mentioned

after

the

presentation

of

the

method

Some

constructions

that

are

only

applicable

to

mixed

covering

codes

are

also

given

Thorough

considerations

of

construction

methods

for

general

mixed

covering

codes

and

binary/ternary

mixed

covering

codes

can

be

found

in

-LSB-

-RSB-

and

-LSB-

-RSB-

respectively

Finally

the

mixed

codes

are

not

only

interesting

in

their

own

right

but

many

times

good

-LRB-

best

known

-RRB-

nonmixed

covering

codes

can

be

constructed

out

of

mixed

covering

codes

see

-LSB-

-RSB-

and

-LSB-

-RSB-

-LRB-

one

useful

theorem

is

given

in

Section

-RRB-

Computer

Search

for

Covering

Codes

The

problem

of

onding

a

-LRB-

q

n

M

-RRB-

R

code

-LRB-

with

q

n

M

and

R

oxed

-RRB-

can

be

seen

as

a

combinatorial

optimization

problem

Such

problems

have

a

onite

set

of

feasible

solutions

and

a

cost

function

that

attaches

a

cost

to

each

such

solution

It

is

now

very

natural

to

let

the

feasible

solutions

be

the

set

of

M-element

subsets

of

Fnq

and

for

a

solution

C

in

this

set

to

set

the

cost

function

to

be

the

number

of

uncovered

words

in

the

space

cost

-LRB-

C

-RRB-

fx

Fnq

j

d

-LRB-

x

C

-RRB-

Rg

If

there

is

a

code

C

such

that

cost

-LRB-

C

-RRB-

then

C

is

a

covering

code

with

the

given

parameters

Now

any

method

for

combinatorial

optimization

can

be

used

to

ond

a

covering

code

-LRB-

see

for

example

-LSB-

-RSB-

-RRB-

To

ond

out

not

only

an

upper

bound

on

Kq

-LRB-

n

R

-RRB-

but

the

exact

value

of

the

function

an

M

must

be

found

such

that

there

is

a

-LRB-

q

n

M

-RRB-

R

code

but

there

is

no

-LRB-

q

n

M

-RRB-

R

code

Whereas

the

proof

of

existence

is

constructive

other

-LRB-

mainly

analytic

-RRB-

methods

have

to

be

used

in

the

nonexistence

proof

Dioeerent

methods

like

linear

programming

-LSB-

-RSB-

have

been

used

in

such

proofs

In

most

cases

the

best

that

can

be

done

is

to

prove

the

nonexistence

of

a

-LRB-

q

n

M

a

-RRB-

R

code

where

a

so

that

there

is

a

gap

between

the

best

known

upper

and

lower

bounds

An

approach

that

is

used

in

-LSB-

-RSB-

to

settle

some

values

of

-LRB-

R

-RRB-

is

now

presented

followed

by

an

approach

which

can

be

used

only

to

give

upper

bounds

A

Set

Covering

Approach

The

problem

of

onding

minimal

covering

codes

can

be

seen

as

a

special

case

of

the

set

covering

problem

-LRB-

SCP

-RRB-

-LSB-

-RSB-

In

the

SCP

an

m?n

zero-one

matrix

aij

and

a

cost

function

cj

associated

with

the

columns

are

given

the

problem

is

now

to

cover

the

rows

by

a

subset

of

columns

at

minimum

cost

In

the

search

for

a

minimal

-LRB-

q

n

M

-RRB-

R

code

a

qn

qn

matrix

is

constructed

where

the

columns

and

the

rows

are

labeled

with

the

words

in

Fnq

-LRB-

in

lexicographic

order

for

example

-RRB-

The

entry

aij

is

if

the

Hamming

distance

between

the

labels

of

the

ith

row

and

the

jth

column

is

less

than

or

equal

to

R

otherwise

The

cost

function

is

cj

it

is

a

unicost

problem

To

ond

out

the

value

of

-LRB-

-RRB-

-LRB-

which

was

already

shown

to

be

-RRB-

the

matrix

in

Figure

is

constructed

Unfortunately

the

facts

that

this

is

a

unicost

problem

that

all

rows

and

columns

have

the

same

number

of

ones

and

that

the

instances

are

in

many

senses

symmetrical

together

collaborate

to

make

the

problem

of

onding

a

minimal

covering

very

hard

Although

many

SCP

instances

can

be

solved

reasonably

fast

this

is

generally

not

the

case

for

covering

codes

However

for

small

covering

codes

this

approach

can

be

used

as

shown

in

-LSB-

-RSB-

where

some

exact

bounds

are

proved

by

computer

search

The

heuristics

of

the

search

is

based

on

ideas

from

-LSB-

-RSB-

where

a

search

for

covering

designs

was

carried

out

In

cases

where

the

exact

value

of

Kq

-LRB-

n

R

-RRB-

can

not

in

reasonable

time

be

found

by

methods

that

solve

the

SCP

eoeorts

can

be

made

to

improve

upper

bounds

on

this

function

by

implicitly

constructing

good

covering

codes

In

the

search

for

such

codes

probabilistic

methods

have

turned

out

to

perform

very

well

Figure

Covering

matrix

for

-LRB-

-RRB-

A

Probabilistic

Approach

In

the

early

's

erny

-LSB-

-RSB-

and

Kirkpatrick

-LSB-

-RSB-

independently

presented

a

probabilistic

method

for

combinatorial

optimization

problems

simulated

annealing

-LRB-

SA

-RRB-

-LSB-

-RSB-

The

orst

results

on

the

use

of

SA

in

the

search

for

codes

are

due

to

El

Gamal

et

al.

-LSB-

-RSB-

They

considered

source

codes

constant

weight

codes

and

spherical

codes

In

Wille

-LSB-

-RSB-

used

SA

to

ond

a

-LRB-

-RRB-

code

thus

proving

that

-LRB-

-RRB-

This

was

a

good

improvement

on

-LRB-

-RRB-

which

was

the

best

known

upper

bound

at

that

time

-LRB-

actually

among

football

pool

constructors

in

the

Nordic

countries

it

was

known

that

-LRB-

-RRB-

which

had

been

proved

by

S.

Solbrand

-RRB-

These

results

encouraged

further

research

on

the

construction

of

covering

codes

by

SA

the

aforementioned

upper

bound

was

later

improved

to

independently

by

Van

Laarhoven

et

al.

-LSB-

-RSB-

and

the

author

of

this

thesis

-LRB-

see

-LSB-

Table

I

-RSB-

-RRB-

SA

has

turned

out

to

be

an

important

tool

in

the

search

for

covering

codes

see

-LSB-

-RSB-

for

a

more

detailed

treatment

of

the

use

of

SA

in

the

search

for

covering

codes

The

-LRB-

-RRB-

code

given

in

-LSB-

-RSB-

was

found

by

SA

Simulated

annealing

has

been

used

to

construct

a

large

number

of

codes

in

the

papers

-LSB-

-RSB-

SA

further

plays

an

important

role

in

some

of

the

methods

to

be

presented

in

the

sequel

Also

as

pointed

out

in

-LSB-

-RSB-

combinatorial

constructions

can

sometimes

be

used

to

obtain

a

code

that

leaves

only

a

small

number

of

words

uncovered

SA

can

then

be

used

to

ond

a

good

covering

of

these

words

As

a

consequence

of

the

probabilistic

nature

of

SA

codes

found

in

repeated

searches

are

not

necessarily

equivalent

The

method

can

thus

be

used

to

ond

nonequivalent

covering

codes

with

given

parameters

whenever

such

exist

Partitioning

Constructions

In

partitioning

constructions

old

codes

are

partitioned

into

subcodes

and

combined

These

constructions

work

eOEciently

whenever

certain

conditions

for

the

partitions

are

fulolled

Such

constructions

are

very

useful

among

many

types

of

codes

for

example

error-correcting

codes

-LSB-

-RSB-

and

constant

weight

codes

-LSB-

-RSB-

it

is

thus

very

natural

to

search

for

analog

methods

for

covering

codes

The

classiocation

of

codes

according

to

properties

of

their

partitions

is

orst

shown

Thereafter

constructions

using

these

codes

are

brieAEy

discussed

For

mixed

covering

codes

the

deonitions

and

constructions

are

essentially

similar

Normal

Seminormal

and

Subnormal

Codes

A

partitioning

construction

for

binary

linear

codes

was

orst

presented

by

Graham

and

Sloane

in

-LSB-

-RSB-

the

amalgamated

direct

sum

-LRB-

ADS

-RRB-

construction

eOEciently

combines

so

called

normal

codes

These

concepts

were

later

generalized

to

the

nonlinear

case

in

-LSB-

-RSB-

In

the

ADS

construction

the

codes

are

partitioned

according

to

the

values

of

given

coordinates

In

-LSB-

-RSB-

Honkala

introduces

subnormal

binary

codes

and

shows

that

normal

binary

codes

consist

of

a

subset

of

subnormal

binary

codes

Deonition

A

-LRB-

n

M

-RRB-

R

code

C

is

subnormal

if

there

are

nonempty

subcodes

and

such

that

C

-LSB-

and

for

all

x

d

-LRB-

x

-RRB-

d

-LRB-

x

-RRB-

Actually

in

the

original

deonition

the

subcodes

are

allowed

to

be

empty

-LSB-

-RSB-

however

since

the

constructions

in

which

subnormal

codes

are

involved

generally

require

the

subcodes

to

be

nonempty

this

requirement

is

included

in

the

deonition

In

-LSB-

-RSB-

Lobstein

and

Van

Wee

generalize

the

concepts

of

normal

codes

and

subnormal

codes

-LRB-

Deonition

-RRB-

to

q-ary

codes

Deonition

A

-LRB-

q

n

M

-RRB-

R

code

C

is

subnormal

if

there

are

nonempty

subcodes

Cq

such

that

C

-LSB-

-LSB-

-LSB-

Cq

and

for

all

x

Fnq

Pni

d

-LRB-

x

Ci

-RRB-

qR

q

The

comment

on

nonemptiness

after

Deonition

also

holds

for

this

deonition

The

ADS

construction

is

easily

generalized

to

the

q-ary

case

as

shown

in

-LSB-

-RSB-

Unfortunately

it

turns

out

that

for

example

no

nonbinary

nontrivial

perfect

codes

are

subnormal

so

it

seems

that

partitioning

constructions

can

not

be

used

in

the

q-ary

case

to

the

same

extent

as

in

the

binary

case

The

following

idea

however

opens

the

door

to

some

new

results

which

show

that

partitioning

constructions

are

of

use

also

in

this

case

The

last

expression

of

Deonition

can

be

written

as

mini

d

-LRB-

x

Ci

-RRB-

maxi

d

-LRB-

x

Ci

-RRB-

leading

to

another

generalization

of

the

same

deonition

Deonition

A

-LRB-

q

n

M

-RRB-

R

code

C

is

subnormal

if

there

are

nonempty

subcodes

Cq

such

that

C

-LSB-

-LSB-

-LSB-

Cq

and

for

all

x

Fnq

mini

d

-LRB-

x

Ci

-RRB-

maxi

d

-LRB-

x

Ci

-RRB-

The

orst

steps

towards

a

generalization

in

this

direction

were

taken

in

-LSB-

Deonition

-RSB-

that

deonition

equals

Deonition

except

that

d

-LRB-

x

C

-RRB-

R

was

used

instead

of

x

Fnq

Such

codes

are

named

seminormal

subnormal

codes

are

apparently

seminormal

but

not

necessarily

vice

versa

Not

only

was

a

partition

of

C

into

q

subcodes

considered

in

-LSB-

-RSB-

more

generally

the

notation

of

k-seminormality

was

used

for

codes

partioned

into

k

-LRB-

arbitrary

-RRB-

rather

than

q

subcodes

The

following

very

general

deonition

is

from

-LSB-

-RSB-

with

the

additional

parameter

introduced

in

-LSB-

-RSB-

-LRB-

In

-LSB-

Section

III

-RSB-

an

even

more

sophisticated

generalization

-LRB-

k

t

-RRB-

subnorm

is

given

however

the

parameter

is

hardly

ever

signiocant

so

it

is

omitted

here

-RRB-

Deonition

A

-LRB-

q

n

M

-RRB-

R

code

C

has

-LRB-

k

t

-RRB-

subnorm

-LRB-

S

-RRB-

if

there

is

a

partition

of

C

into

k

nonempty

subsets

Ck

such

that

mina

d

-LRB-

x

Ca

-RRB-

maxa

d

-LRB-

x

Ca

-RRB-

S

whenever

R

t

d

-LRB-

x

C

-RRB-

R

and

for

all

x

maxa

d

-LRB-

x

Ca

-RRB-

Such

a

partition

is

called

acceptable

A

code

C

is

said

to

be

-LRB-

k

t

-RRB-

subnormal

if

it

has

-LRB-

k

t

-RRB-

subnorm

-LRB-

-RRB-

-LRB-

for

some

the

parameter

is

then

usually

omitted

-RRB-

Thus

-LRB-

k

-RRB-

subnormality

coincides

with

k-seminormality

Furthermore

a

code

is

called

strongly

seminormal

-LSB-

-RSB-

if

it

has

-LRB-

k

R

-RRB-

subnorm

-LRB-

R

-RRB-

An

interesting

question

is

whether

the

parameter

t

in

Deonition

is

signiocant

It

seems

that

there

are

very

few?especially

optimal

codes

that

have

-LRB-

k

a

-RRB-

subnorm

S

but

that

do

not

have

-LRB-

k

b

-RRB-

subnorm

S

a

b

A

necessary

condition

for

this

to

be

true

is

that

a

b.

Altogether

such

codes

exist

The

-LRB-

-RRB-

repetition

code

has

-LRB-

-RRB-

subnorm

but

has

-LRB-

-RRB-

subnorm

S

only

for

S

An

algorithm

using

simulated

annealing

for

onding

partitions

of

k-seminormal

codes

is

given

in

-LSB-

Section

IV

-RSB-

The

concept

of

normality

was

mentioned

very

brieAEy

earlier

Although

generally

not

as

important

as

subnormality

in

the

construction

of

nonlinear

covering

codes

normality

plays

an

important

role

in

some

eOEcient

constructions

for

example

in

-LSB-

Corollary

-RSB-

The

deonition

of

-LRB-

k

t

-RRB-

norm

is

here

stated

as

in

-LSB-

Deonition

-RSB-

Let

C

be

a

-LRB-

q

n

M

-RRB-

R

code

with

C

-LRB-

i

-RRB-

a

f

-LRB-

ci

ci

cn

-RRB-

j

-LRB-

ci

a

ci

cn

-RRB-

Cg

a

Fq

These

sets

are

assumed

to

be

nonempty

for

all

a.

Deonition

The

code

C

has

-LRB-

q

t

-RRB-

norm

-LRB-

N

-LRB-

i

-RRB-

-RRB-

with

respect

to

the

ith

coordinate

if

mina

d

-LRB-

x

C

-LRB-

i

-RRB-

a

-RRB-

maxa

d

-LRB-

x

C

-LRB-

i

-RRB-

a

-RRB-

N

-LRB-

i

-RRB-

whenever

R

t

d

-LRB-

x

S

a

C

-LRB-

i

-RRB-

a

-RRB-

R

and

for

all

x

maxa

d

-LRB-

x

C

-LRB-

i

-RRB-

a

-RRB-

If

N

-LRB-

i

-RRB-

N

for

at

least

one

coordinate

i

C

is

said

to

have

-LRB-

q

t

-RRB-

norm

-LRB-

N

-RRB-

and

coordinates

for

which

this

equation

holds

are

called

acceptable

A

code

C

is

said

to

be

-LRB-

q

t

-RRB-

normal

if

it

has

-LRB-

q

t

-RRB-

norm

-LRB-

the

value

of

is

here

not

essential

-RRB-

As

can

be

seen

the

orst

parameter

-LRB-

k

-RRB-

in

the

deonition

of

-LRB-

k

t

-RRB-

norm

is

superAEuous

in

the

nonmixed

case

as

it

is

always

q

however

in

the

mixed

case

its

presence

is

justioed

A

lot

of

research

has

been

done?mainly

on

binary

covering

codes?to

determine

normality

and

subnormality

properties

of

-LRB-

optimal

and

other

-RRB-

covering

codes

-LSB-

-RSB-

Also

as

pointed

out

in

-LSB-

-RSB-

some

early

results

by

Golomb

and

Posner

-LSB-

-RSB-

actually

deal

with

partitions

of

covering

codes

Combining

Codes

Codes

with

the

aforementioned

properties?normality

seminormality

and

subnormality?can

now

eOEciently

be

combined

to

obtain

record-breaking

codes

The

constructions

act

on

codes

say

A

and

B

that

are

partitioned

into

blocks

of

codewords

A

-LSB-

-LSB-

-LSB-

Ak

and

B

-LSB-

-LSB-

-LSB-

Bk

For

all

i

all

words

in

Ai

and

Bi

are

combined

to

get

C

-LSB-

ki

-LRB-

Ai

Bi

-RRB-

Now

if

the

aforementioned

partitions

are

acceptable

partitions

corresponding

to

-LRB-

k

t

-RRB-

-LRB-

sub

-RRB-

norms

-LRB-

-RRB-

and

-LRB-

NB

-RRB-

an

upper

bound

on

the

covering

radius

of

C

can

be

calculated

by

combinatorial

arguments

It

turns

out

that

especially

-LRB-

q

R

-RRB-

normal

-LSB-

Corollary

-RSB-

seminormal

-LSB-

Theorem

-RSB-

strongly

seminormal

-LSB-

Theorem

-RSB-

and

-LRB-

q

R

-RRB-

subnormal

-LSB-

Theorem

-RSB-

covering

codes

are

important

building

blocks

In

-LSB-

-RSB-

Honkala

names

this

construction

blockwise

direct

sum

-LRB-

BDS

-RRB-

The

ADS

construction

mentioned

earlier

is

a

BDS

construction

where

at

least

one

of

the

codes

involved

is

partitioned

according

to

the

values

of

one

of

its

coordinates

-LRB-

cf.

-LRB-

k

t

-RRB-

norm

-RRB-

By

using

cosets

of

a

Hamming

code

as

one

of

the

codes

that

are

combined

the

following

very

useful

theorem

can

be

obtained

-LSB-

Theorem

-RSB-

Theorem

If

q

is

a

prime

power

and

k

then

q

qk

qm

-LRB-

-LRB-

qk

-RRB-

-LRB-

q

-RRB-

nm

R

-RRB-

q

-LRB-

qk

-RRB-

-LRB-

q

-RRB-

k

q

qk

qm

-LRB-

nm

R

-RRB-

The

theorems

in

-LSB-

-RSB-

are

special

instances

of

this

theorem

Similar

constructions

are

also

discussed

in

-LSB-

-RSB-

The

partitioning

constructions

and

the

deonitions

of

dioeerent

degrees

of

normality

given

here

can

without

diOEculties

be

carried

over

to

the

mixed

case

details

can

be

found

in

-LSB-

-RSB-

A

Matrix

Method

The

possibility

of

using

computers

in

the

search

for

covering

codes

was

discussed

in

Section

Simulated

annealing

works

very

well

as

long

as

the

codes

are

relatively

small

-LRB-

for

example

up

to

about

codewords

for

R

-RRB-

but

the

performance

deteriorates

quite

fast

with

an

increasing

number

of

codewords

One

solution

to

this

problem

is

to

search

for

a

code

with

some

kind

of

structure

imposed

on

it

A

search

could

be

carried

out

for

for

example

a

self-complementary

code

A

binary

code

is

self-complementary

if

c

is

a

codeword

whenever

c

is

a

codeword

The

code

in

Figure

has

this

property

The

requirement

can

be

rewritten

in

the

following

way

c

is

a

codeword

whenever

c

is

a

codeword

so

the

code

consists

of

cosets

of

the

linear

code

By

using

the

matrix

method

codes

that

consist

of

cosets

of

a

linear

code

can

be

found

The

orst

ideas

that

led

to

this

method

were

presented

by

Kamps

and

Van

Lint

-LSB-

-RSB-

in

these

ideas

were

later

reoned

by

Blokhuis

and

Lam

-LSB-

-RSB-

in

Blokhuis

and

Lam

considered

the

construction

of

q-ary

codes

with

covering

radius

The

method

is

easily

generalized

to

many

cases

as

shown

in

-LSB-

-RSB-

-LRB-

arbitrary

covering

radius

-RRB-

-LSB-

-RSB-

-LRB-

dioeerent

kinds

of

multiple

coverings

-RRB-

and

-LSB-

-RSB-

-LRB-

mixed

codes

-RRB-

For

arbitrary

covering

radius

the

following

holds

Let

A

-LRB-

I

M

-RRB-

-LRB-

an

-RRB-

be

a

r

n

matrix

where

I

is

the

r

r

identity

matrix

and

M

is

an

r

-LRB-

n

r

-RRB-

matrix

with

entries

from

Fq

Now

for

s

F

rq

SA

R

-LRB-

s

-RRB-

fs

n

X

j

jaj

j

j

Fq

jfj

j

j

gj

Rg

and

s

is

said

to

R-cover

SA

R

-LRB-

s

-RRB-

using

A.

Consequently

A

I

corresponds

to

covering

in

the

traditional

sense

A

subset

S

of

F

rq

is

said

to

R-cover

F

rq

using

A

if

F

rq

-LSB-

SA

R

-LRB-

s

-RRB-

Theorem

-LRB-

Carnielli

-LSB-

Theorem

-RSB-

Van

Lint

Jr.

-LSB-

Theorem

-RSB-

-RRB-

If

S

R-covers

F

rq

using

an

r

n

matrix

A

-LRB-

I

M

-RRB-

then

W

fw

Fnq

j

Aw

Sg

covers

Fnq

with

radius

R.

jW

j

jSj

qn?r

It

is

far

from

obvious

how

to

make

use

of

this

theorem

in

practice

First

when

attempts

are

made

to

improve

a

certain

bound

Kq

-LRB-

n

R

-RRB-

M

the

value

of

r

must

be

oxed

After

that

both

the

elements

of

the

set

S

and

a

matrix

M

that

lead

to

a

covering

have

to

be

found

In

-LSB-

-RSB-

the

method

is

used

to

prove

that

-LRB-

-RRB-

The

heuristics

is

not

explained

the

authors

only

mention

that

the

code

was

found

in

a

computer

search

In

-LSB-

-RSB-

simulated

annealing

is

used

during

the

search

both

the

words

in

S

and

the

columns

of

A

are

altered

By

using

this

approach

a

recordbreaking

-LRB-

-RRB-

code

was

found

In

-LSB-

-RSB-

the

fact

that

the

method

gives

us

a

code

that

consists

of

cosets

of

a

linear

code

is

utilized

Then

all

nonequivalent

linear

codes

-LSB-

-RSB-

with

given

parameters

can

be

considered

and

a

search

can

be

carried

out

by

simulated

annealing

with

the

matrix

A

oxed

to

parity

check

matrices

for

these

codes

one

at

a

time

Unfortunately

the

problem

of

onding

all

such

nonequivalent

codes

is

a

very

hard

problem

The

following

restriction

is

then

made

only

linear

codes

with

dioeerent

weight

distributions

are

considered

This

invariant

is

quite

sensitive?all

nonequivalent

codes

can

not

be

distinguished

but

most

of

them?and

the

codes

can

be

found

quite

eOEciently

The

results

also

encourage

the

approach

two

new

upper

bounds

for

the

football

pool

problem

were

found

in

-LSB-

-RSB-

and

many

new

binary/ternary

covering

codes

were

found

in

-LSB-

-RSB-

It

is

further

interesting

to

see

that

the

linear

code

used

in

the

proof

of

-LRB-

-RRB-

in

-LSB-

-RSB-

has

the

same

length

and

the

same

number

of

codewords

as

the

-LRB-

-RRB-

Golay

code

however

these

are

not

equivalent

Thus

cosets

of

good

linear

codes

-LRB-

small

covering

radius

-RRB-

do

not

necessarily

lead

to

the

most

eOEcient

coverings

The

-LRB-

-RRB-

code

given

in

-LSB-

-RSB-

was

actually

found

by

this

method

It

consists

of

a

union

of

cosets

of

a

linear

code

If

-LRB-

a

b

c

d

e

-RRB-

is

a

codeword

in

the

code

so

is

-LRB-

a

b

c

;d

e

-RRB-

-LRB-

mod

-RRB-

The

bound

-LRB-

-RRB-

which

was

proved

in

-LSB-

-RSB-

can

also

be

proved

by

the

matrix

method

With

jSj

r

and

n

coverings

can

be

found

with

M

-LRB-

-RRB-

T

and

M

-LRB-

-RRB-

T

Actually

the

second

code

in

-LSB-

Fig.

-RSB-

consists

of

cosets

of

a

linear

code

In

the

search

for

-LRB-

-RRB-

codes

in

-LSB-

-RSB-

approximately

h

of

CPU

time

on

a

VAX

computer

was

used

to

ond

two

codes

By

the

matrix

method

codes

are

found

considerably

faster

In

the

mixed

case

as

explained

in

-LSB-

-RSB-

care

must

be

taken

when

choosing

the

matrices

Arities

are

associated

with

the

rows

and

the

columns

of

A

and

an

entry

of

A

can

have

a

nonzero

value

of

Fq

only

if

the

arities

associated

with

the

row

and

the

column

of

the

entry

both

are

q.

Other

Combinatorial

Methods

A

wide

variety

of

combinatorial

construction

methods

has

been

published

through

the

years

In

this

section

methods

that

can

be

generally

applied

and

bounds

obtained

by

such

methods

are

presented

Some

Direct

Constructions

General

upper

bounds

corresponding

to

constructions

of

good

q-ary

covering

codes

with

n

R

large

are

orst

presented

Theorem

is

trivial

Theorems

and

are

given

in

-LSB-

Theorem

-RSB-

and

-LSB-

Theorem

-RSB-

respectively

Theorem

Kq

-LRB-

n

R

-RRB-

if

n

R.

Theorem

Kq

-LRB-

n

R

-RRB-

q

if

R

n

R

dn

qe

Theorem

Kq

-LRB-

qr

p

qr

r

p

-RRB-

q

-LRB-

p

-RRB-

if

p

q

See

-LSB-

-RSB-

for

corresponding

bounds

for

mixed

codes

Some

upper

bounds

that

correspond

to

constructions

that

can

be

applied

directly

to

existing

covering

codes

are

now

given

Theorem

can

not

be

generalized

to

the

mixed

case

whereas

Theorem

explicitly

concerns

mixed

codes

Theorem

Kq

-LRB-

n

R

-RRB-

qKq

-LRB-

n

R

-RRB-

Theorem

Kq

-LRB-

-RRB-

Kq

-LRB-

-RRB-

Kq

-LRB-

-RRB-

Theorem

Kqs

-LRB-

n

R

-RRB-

sn

-LRB-

n

R

-RRB-

Theorem

qm

-LRB-

nm

R

-RRB-

-LRB-

-RRB-

qm

-LRB-

nm

R

-RRB-

if

Theorems

and

are

trivial

Theorem

is

obtained

by

combining

codes

using

the

direct

sum

construction

-LSB-

p.

-RSB-

and

Theorem

can

be

obtained

as

a

special

case

of

Theorem

-LRB-

see

-LSB-

Theorem

-RSB-

-RRB-

Theorem

is

mentioned

in

the

seminal

paper

of

Taussky

and

Todd

-LSB-

-RSB-

-LRB-

for

R

-RRB-

Theorem

is

proved

in

-LSB-

-RSB-

-LRB-

for

R

-RRB-

Theorem

and

a

slight

modiocation

of

it

are

discussed

in

-LSB-

-RSB-

A

good

upper

bound

is

obtained

by

Theorem

in

cases

where

the

bound

Kq

-LRB-

n

R

-RRB-

is

good

starting

from

a

q-ary

Hamming

code

of

length

q

inequality

can

be

replaced

by

equality

in

Theorem

By

a

construction

similar

to

that

in

the

proof

of

Theorem

-LRB-

-LSB-

Theorem

-RSB-

-RRB-

one

can?starting

from

the

aforementioned

Hamming

codes?not

only

get

good

codes

of

arity

qs

but

of

any

arity

greater

than

q

-LSB-

-RSB-

Piecewise

Constant

Codes

The

concept

of

piecewise

constant

codes

was

introduced

in

-LSB-

-RSB-

The

coordinates

of

such

a

code

are

partitioned

into

t

blocks

Now

if

c

is

a

codeword

then

all

words

that

can

be

obtained

by

all

possible

permutations

among

the

coordinates

of

one

block

are

also

codewords

The

following

code

is

an

example

of

a

-LRB-

-RRB-

piecewise

constant

code

-LRB-

with

t

-RRB-

In

-LSB-

-RSB-

a

useful

graphical

method

for

determining

the

covering

radius

of

a

piecewise

constant

code

is

presented

The

fact

that

permutations

of

coordinates

leave

the

code

unchanged

means

that

the

code

has

a

nontrivial

automorphism

group

An

automorphism

group

of

the

aforementioned

-LRB-

-RRB-

code

can

be

generated

by

the

permutations

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

the

order

of

the

group

is

The

code

is

obtained

by

applying

these

permutations

to

for

example

the

following

set

of

words

It

is

interesting

to

see

that

the

code

can

be

generated

by

applying

the

group

generated

by

-LRB-

-RRB-

-LRB-

-RRB-

to

these

words

the

order

of

this

group

is

only

The

other

codes

in

-LSB-

-RSB-

that

are

given

as

constant

weight

codes

can

also

be

described

nicely

by

a

single

generator

The

-LRB-

-RRB-

code

in

-LSB-

Fig.

-RSB-

is

obtained

by

applying

the

group

generated

by

-LRB-

-RRB-

-LRB-

-RRB-

to

The

-LRB-

-RRB-

code

in

-LSB-

Fig.

-RSB-

can

be

constructed

so

that

-LRB-

-RRB-

-LRB-

-RRB-

is

an

automorphism

group

The

crucial

part

is

that

there

is

a

Steiner

system

S

-LRB-

-RRB-

with

the

aforementioned

automorphism

group

-LSB-

-RSB-

Apparently

no

exhaustive

study

of

the

use

of

automorphism

groups

in

the

search

for

good

covering

codes

has

been

performed

The

examples

presented

here

encourage

research

in

this

area

this

method

could

possibly

be

a

good

alternative

to

the

matrix

method

in

the

search

for

large

codes

Lengthening

Constructions

It

is

easily

shown

that

if

C

is

a

-LRB-

q

n

M

-RRB-

R

code

-LRB-

q

is

a

prime

power

-RRB-

then

C

R-covers

Fn

q

using

the

-LRB-

n

-RRB-

-LRB-

qn

-RRB-

matrix

A

q

q

All

unmarked

entries

are

zeroes

From

the

results

presented

in

Section

the

following

lengthening

theorem

is

then

obtained

Theorem

If

q

is

a

prime

power

then

Kq

-LRB-

qn

R

-RRB-

qn

-LRB-

q

-RRB-

Kq

-LRB-

n

R

-RRB-

For

q

a

prime

and

R

this

theorem

was

proved

in

-LSB-

-RSB-

it

was

later

generalized

to

prime

powers

and

arbitrary

radii

in

-LSB-

-RSB-

In

-LSB-

-RSB-

a

proof

of

the

case

q

R

is

given

based

on

another

construction

found

by

Mollard

and

independently

Katsman

and

Litsyn

that

construction

can

also

be

generalized

in

another

way

as

shown

by

Honkala

and

H?m?l?inen

-LSB-

-RSB-

This

is

a

very

useful

construction

for

R

For

example

for

a

oxed

q

starting

from

the

Hamming

code

of

length

q

a

series

of

codes

is

obtained

with

the

same

parameters

as

the

q-ary

Hamming

codes

by

repeatedly

applying

the

construction

that

proves

Theorem

Unfortunately

the

construction

can

not

easily

be

generalized

to

arbitrary

covering

radii

In

-LSB-

-RSB-

however

Honkala

uses

some

results

about

onite

oelds

to

develop

a

lengthening

construction

that

works

eOEciently

in

many

cases

where

R

generalizing

many

earlier

results

including

Theorem

for

q

a

prime

and

some

results

in

-LSB-

-RSB-

The

construction

proves

the

following

theorem

Theorem

If

q

is

a

prime

then

Kq

-LRB-

tn

R

-RRB-

sq

-LRB-

t

-RRB-

nKq

-LRB-

n

R

-RRB-

where

s

q

-LRB-

t

-RRB-

Rg

Upper

bounds

that

this

theorem

leads

to

can

be

found

in

-LSB-

-RSB-

For

q

a

prime

power

Honkala

gives

similar

constructions

in

-LSB-

-RSB-

These

lengthening

constructions

can

not

be

eOEciently

generalized

to

codes

over

mixed

alphabets

Conclusions

In

this

thesis

construction

methods

-LRB-

occasionally

in

the

form

of

upper

bounds

that

are

proved

by

the

constructions

-RRB-

for

covering

codes

are

presented

The

new

results

in

-LSB-

-RSB-

to

-LSB-

-RSB-

are

surveyed

and

related

to

earlier

results

on

this

problem

The

use

of

computers

has

played

a

central

role

in

the

research

many

of

the

methods

presented

are

based

on

computer

search

Simulated

annealing

has

turned

out

to

perform

very

well

in

many

cases

Although

a

new

code

that

is

found

in

such

a

search

is

not

as

valuable

as

if

it

were

proved

mathematically

such

proofs

can

many

times

be

found

by

investigating

the

code

found

in

the

search

Examples

of

this

are

the

ternary

covering

codes

of

length

and

that

are

treated

in

-LSB-

-RSB-

and

-LSB-

-RSB-

respectively

In

all

papers

-LSB-

-RSB-

to

-LSB-

-RSB-

not

only

construction

methods

are

presented

but

also

record-breaking

upper

bounds

that

these

methods

lead

to

The

following

extensive

tables

of

upper

bounds

can

be

found

in

the

publications

-LSB-

-RSB-

-LRB-

n

R

-RRB-

for

n

R

-LRB-

n

R

-RRB-

for

n

R

and

-LRB-

n

R

-RRB-

for

n

R

-LSB-

-RSB-

-LRB-

R

-RRB-

for

R

-LSB-

-RSB-

-LRB-

n

-RRB-

for

n

and

-LSB-

-RSB-

-LRB-

R

-RRB-

for

R

As

for

applications

of

covering

codes

the

possibility

of

using

them

as

football

pool

systems

was

already

mentioned

this

is

also

the

main

application

Other

possible

applications

include

distribution

of

resources

in

hypercube

computers

-LSB-

-RSB-

and

coding

of

speech

-LSB-

-RSB-

A

number

of

other

applications

of

covering

codes

in

dioeerent

problems

of

information

theory

and

computer

science

are

mentioned

for

example

in

-LSB-

-RSB-

References

-LSB-

-RSB-

E.

Aarts

and

J.

Korst

Simulated

Annealing

and

Boltzmann

Machines

A

Stochastic

Approach

to

Combinatorial

Optimization

and

Neural

Computing

Wiley

Chichester

-LSB-

-RSB-

E.

Balas

and

A.

Ho

Set

covering

algorithms

using

cutting

planes

heuristics

and

subgradient

optimization

a

computational

study

Math

Programming

-LRB-

-RRB-

-LSB-

-RSB-

J.

E.

Beasley

An

algorithm

for

set

covering

problem

European

J.

Oper

Res

-LRB-

-RRB-

-LSB-

-RSB-

A.

Blokhuis

and

C.

W.

H.

Lam

More

coverings

by

rook

domains

J.

Combin

Theory

Ser

A

-LRB-

-RRB-

-LSB-

-RSB-

A.

E.

Brouwer

J.

B.

Shearer

N.

J.

A.

Sloane

and

W.

D.

Smith

A

new

table

of

constant

weight

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

R.

A.

Brualdi

V.

S.

Pless

and

R.

M.

Wilson

Short

codes

with

a

given

covering

radius

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

R.

A.

Brualdi

and

V.

S.

Pless

On

the

length

of

codes

with

a

given

covering

radius

in

D.

Ray-Chauduri

-LRB-

ed

-RRB-

Coding

Theory

and

Design

Theory

Part

I

Coding

Theory

Springer-Verlag

New

York

pp

-LSB-

-RSB-

A.

R.

Calderbank

and

N.

J.

A.

Sloane

Inequalities

for

covering

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

W.

A.

Carnielli

On

covering

and

coloring

problems

for

rook

domains

Discrete

Math

-LRB-

-RRB-

-LSB-

-RSB-

W.

A.

Carnielli

Hyper-rook

domain

inequalities

Stud

Appl

Math

-LRB-

-RRB-

-LSB-

-RSB-

V.

erny

Thermodynamical

approach

to

the

traveling

salesman

problem

an

eOEcient

simulation

algorithm

J.

Opt

Theory

Appl

-LRB-

-RRB-

-LSB-

-RSB-

W.

Chen

and

I.

S.

Honkala

Lower

bounds

for

q-ary

covering

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

N.

Christoodes

and

S.

Korman

A

computational

survey

of

methods

for

the

set

covering

problem

Management

Sci

-LRB-

-RRB-

-LSB-

-RSB-

R.

F.

Clayton

Multiple

Packings

and

Coverings

in

Algebraic

Coding

Theory

Ph.D.

thesis

Univ.

of

California

Los

Angeles

-LSB-

-RSB-

G.

D.

Cohen

M.

G.

Karpovsky

H.

F.

Mattson

Jr.

and

J.

R.

Schatz

Covering

radius?survey

and

recent

results

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

G.

D.

Cohen

A.

C.

Lobstein

and

N.

J.

A.

Sloane

Further

results

on

the

covering

radius

of

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

T.

J.

Dickson

On

a

covering

problem

concerning

Abelian

groups

J.

London

Math

Soc

-LRB-

-RRB-

-LRB-

-RRB-

-LSB-

-RSB-

A.

A.

El

Gamal

L.

A.

Hemachandra

I.

Shperling

and

V.

K.

Wei

Using

simulated

annealing

to

design

good

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

J.

Etcheberry

The

set-covering

problem

a

new

implicit

enumeration

algorithm

Oper

Res

-LRB-

-RRB-

-LSB-

-RSB-

T.

Etzion

and

G.

Greenberg

Constructions

for

perfect

mixed

codes

and

other

covering

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

H.

Fernandes

and

E.

Rechtschaoeen

The

football

pool

problem

for

and

matches

J.

Combin

Theory

Ser

A

-LRB-

-RRB-

-LSB-

-RSB-

A.

B.

Fontaine

and

W.

W.

Peterson

Group

code

equivalence

and

optimum

codes

IEEE

Trans

Inform

Theory

-LRB-

Special

Supplement

-RRB-

-LRB-

-RRB-

-LSB-

-RSB-

S.

W.

Golomb

and

E.

C.

Posner

Rook

domains

Latin

squares

aOEne

planes

and

error-distributing

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

R.

L.

Graham

and

N.

J.

A.

Sloane

On

the

covering

radius

of

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

H.

O.

H?m?l?inen

I.

S.

Honkala

M.

K.

Kaikkonen

and

S.

N.

Litsyn

Bounds

for

binary

multiple

covering

codes

Des

Codes

Cryptogr

-LRB-

-RRB-

-LSB-

-RSB-

H.

O.

H?m?l?inen

I.

S.

Honkala

and

S.

N.

Litsyn

Bounds

for

binary

codes

that

are

multiple

coverings

of

the

farthest-ooe

points

submitted

for

publication

-LSB-

-RSB-

H.

H?m?l?inen

and

S.

Rankinen

Upper

bounds

for

football

pool

problems

and

mixed

covering

codes

J.

Combin

Theory

Ser

A

-LRB-

-RRB-

-LSB-

-RSB-

O.

Heden

A

generalized

Lloyd

theorem

and

mixed

perfect

codes

Mathematica

Scand

-LRB-

-RRB-

-LSB-

-RSB-

O.

Heden

A

new

construction

of

group

and

nongroup

perfect

codes

Inform

and

Control

-LRB-

-RRB-

-LSB-

-RSB-

M.

Herzog

and

J.

Sch?nheim

Group

partition

factorization

and

the

vector

covering

problem

Canad

Math

Bull

-LRB-

-RRB-

-LSB-

-RSB-

M.

Herzog

and

J.

Sch?nheim

Linear

and

nonlinear

single-error-correcting

perfect

mixed

codes

Inform

and

Control

-LRB-

-RRB-

-LSB-

-RSB-

I.

S.

Honkala

Lower

bounds

for

binary

covering

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

I.

S.

Honkala

On

the

normality

of

codes

with

covering

radius

one

in

Proc

Fourth

Joint

Swedish-Soviet

Int

Workshop

Inform

Theory

Studentlitteratur

Lund

pp.

-LSB-

-RSB-

I.

S.

Honkala

Modioed

bounds

for

covering

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

I.

S.

Honkala

On

-LRB-

k

t

-RRB-

subnormal

covering

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

I.

S.

Honkala

All

binary

codes

with

covering

radius

one

are

subnormal

Discrete

Math

-LRB-

-RRB-

-LSB-

-RSB-

I.

S.

Honkala

On

lengthening

of

covering

codes

Discrete

Math

-LRB-

-RRB-

-LSB-

-RSB-

I.

Honkala

On

-LRB-

q

-RRB-

subnormal

q-ary

covering

codes

Discrete

Appl

Math.

to

appear

-LSB-

-RSB-

I.

S.

Honkala

and

H.

O.

H?m?l?inen

A

new

construction

for

covering

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

I.

S.

Honkala

and

H.

O.

H?m?l?inen

Bounds

for

abnormal

binary

codes

with

covering

radius

one

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

X.

Hou

Some

results

on

the

norm

of

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

X.

Hou

New

lower

bounds

for

covering

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

X.

Hou

An

improved

sphere

covering

bound

for

the

codes

with

n

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

X.

Hou

Binary

linear

quasi-perfect

codes

are

normal

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

G.

A.

Kabatyanskii

and

V.

I.

Panchenko

Unit

sphere

packings

and

coverings

of

the

Hamming

space

Problemy

Peredachi

Informatsii

-LRB-

-RRB-

-LRB-

in

Russian

English

translation

in

Problems

Inform

Transmission

-LRB-

-RRB-

-RRB-

-LSB-

-RSB-

J.

G.

KalbAEeisch

and

R.

G.

Stanton

A

combinatorial

problem

in

matching

J.

London

Math

Soc

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

-LRB-

-RRB-

-LSB-

-RSB-

J.

G.

KalbAEeisch

R.

G.

Stanton

and

J.

D.

Horton

On

covering

sets

and

error-correcting

codes

J.

Combin

Theory

-LRB-

-RRB-

-LSB-

-RSB-

J.

G.

KalbAEeisch

and

P.

H.

Weiland

Some

new

results

for

the

covering

problem

in

W.

T.

Tutte

-LRB-

ed

-RRB-

Recent

Progress

in

Combinatorics

Academic

Press

New

York

pp.

-LSB-

-RSB-

H.

J.

L.

Kamps

and

J.

H.

van

Lint

The

football

pool

problem

for

matches

J.

Combin

Theory

-LRB-

-RRB-

-LSB-

-RSB-

H.

J.

L.

Kamps

and

J.

H.

van

Lint

A

covering

problem

in

Colloq

Math

Soc

J?nos

Bolyai

Hung

Combin

Theory

and

Appl.

Balantonf?red

Hungary

pp.

-LSB-

-RSB-

K.

E.

Kilby

and

N.

J.

A.

Sloane

On

the

covering

radius

problem

for

codes

I.

Bounds

on

normalized

covering

radius

SIAM

J.

Alg

Discr

Meth

-LRB-

-RRB-

-LSB-

-RSB-

K.

E.

Kilby

and

N.

J.

A.

Sloane

On

the

covering

radius

problem

for

codes

II

Codes

of

low

dimension

normal

and

abnormal

codes

SIAM

J.

Alg

Discr

Meth

-LRB-

-RRB-

-LSB-

-RSB-

S.

Kirkpatrick

C.

D.

Gelatt

Jr.

and

M.

P.

Vecchi

Optimization

by

simulated

annealing

Science

-LRB-

-RRB-

-LSB-

-RSB-

E.

Kolev

Codes

over

GF

-LRB-

-RRB-

of

length

codewords

and

covering

radius

submitted

for

publication

-LSB-

-RSB-

P.

J.

M.

van

Laarhoven

and

E.

H.

L.

Aarts

Simulated

Annealing

Theory

and

Applications

Reidel

Dordrecht

-LSB-

-RSB-

P.

J.

M.

van

Laarhoven

E.

H.

L.

Aarts

J.

H.

van

Lint

and

L.

T.

Wille

New

upper

bounds

for

the

football

pool

problem

for

and

matches

J.

Combin

Theory

Ser

A

-LRB-

-RRB-

-LSB-

-RSB-

H.

W.

Lenstra

Jr.

Two

theorems

on

perfect

codes

Discrete

Math

-LRB-

-RRB-

-LSB-

-RSB-

D.

Li

and

W.

Chen

New

lower

bounds

for

binary

covering

codes

submitted

for

publication

-LSB-

-RSB-

B.

Lindstr?m

Group

partitions

and

mixed

perfect

codes

Canad

Math

Bull

-LRB-

-RRB-

-LSB-

-RSB-

J.

H.

van

Lint

A

survey

of

perfect

codes

Rocky

Mountain

J.

Math

-LRB-

-RRB-

-LSB-

-RSB-

J.

H.

van

Lint

Introduction

to

Coding

Theory

Springer-Verlag

New

York

-LSB-

-RSB-

J.

H.

van

Lint

Recent

results

on

covering

problems

in

T.

Mora

-LRB-

ed

-RRB-

Applied

Algebra

Algebraic

Algorithms

and

Error-Correcting

Codes

LNCS

Springer-Verlag

Berlin

pp.

-LSB-

-RSB-

J.

H.

van

Lint

Jr.

Covering

Radius

Problems

M.Sc

thesis

Eindhoven

University

of

Technology

The

Netherlands

June

-LSB-

-RSB-

J.

H.

van

Lint

Jr.

and

G.

J.

M.

van

Wee

Generalized

bounds

on

binary/ternary

mixed

packing

and

covering

codes

J.

Combin

Theory

Ser

A

-LRB-

-RRB-

-LSB-

-RSB-

M.

Livingston

and

Q.F.

Stout

Distributing

resources

in

hypercube

computers

in

G.

Fox

-LRB-

ed

-RRB-

Proc

of

the

Third

Conference

on

Hypercube

Multiprocessors

Pasadena

pp.

-LSB-

-RSB-

A.

C.

Lobstein

and

G.

J.

M.

van

Wee

On

normal

and

subnormal

q-ary

covering

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

and

-LRB-

-RRB-

-LSB-

-RSB-

G.

O.

Losey

Note

on

a

theorem

of

Zaremba

J.

Combin

Theory

-LRB-

-RRB-

-LSB-

-RSB-

F.

J.

MacWilliams

and

N.

J.

A.

Sloane

The

Theory

of

Error-Correcting

Codes

North-Holland

Amsterdam

-LSB-

-RSB-

E.

Mattioli

Sopra

una

particolare

proprieta

dei

gruppi

abeliani

oniti

Ann

Scuola

Norm

Sup

Pisa

-LRB-

-RRB-

-LRB-

-RRB-

-LSB-

-RSB-

J.

G.

Mauldon

Covering

theorems

for

groups

Quart

J.

Math

Oxford

-LRB-

-RRB-

-LRB-

-RRB-

-LSB-

-RSB-

V.

I.

Panchenko

Packings

and

coverings

over

an

arbitrary

alphabet

Problemy

Peredachi

Informatsii

-LRB-

-RRB-

-LRB-

in

Russian

English

translation

in

Problems

Inform

Transmission

-LRB-

-RRB-

-RRB-

-LSB-

-RSB-

C.

H.

Papadimitriou

and

K.

Steiglitz

Combinatorial

Optimization

Algorithms

and

Complexity

Prentice-Hall

New

York

-LSB-

-RSB-

K.

T.

Phelps

A

general

product

construction

for

error

correcting

codes

SIAM

J.

Alg

Discr

Meth

-LRB-

-RRB-

-LSB-

-RSB-

V.

Pless

Introduction

to

the

Theory

of

Error-Correcting

Codes

Wiley

New

York

-LSB-

-RSB-

C.

L.

M.

van

Pul

and

T.

Etzion

New

lower

bounds

for

constant

weight

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

E.

R.

Rodemich

Coverings

by

rook

domains

J.

Combin

Theory

-LRB-

-RRB-

-LSB-

-RSB-

D.

Slepian

Some

further

theory

of

group

codes

Bell

System

Tech

J.

-LRB-

-RRB-

-LSB-

-RSB-

N.

J.

A.

Sloane

A

new

approach

to

the

covering

radius

of

codes

J.

Combin

Theory

Ser

A

-LRB-

-RRB-

-LSB-

-RSB-

N.

J.

A.

Sloane

S.

M.

Reddy

and

C.

L.

Chen

New

binary

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

R.

G.

Stanton

Covering

theorems

in

groups

-LRB-

or

How

to

win

at

football

pools

-RRB-

in

W.

T.

Tutte

-LRB-

ed

-RRB-

Recent

Progress

in

Combinatorics

Academic

Press

New

York

pp.

-LSB-

-RSB-

R.

G.

Stanton

and

J.

A.

Bate

A

computer

search

for

B-coverings

in

R.

W.

Robinson

G.

W.

Southern

and

W.

D.

Wallis

-LRB-

eds

-RRB-

Combinatorial

Mathematics

VII

LNM

Springer-Verlag

Berlin

pp.

-LSB-

-RSB-

R.

G.

Stanton

J.

D.

Horton

and

J.

G.

KalbAEeisch

Covering

theorems

for

vectors

with

special

reference

to

the

case

of

four

and

ove

components

J.

London

Math

Soc

-LRB-

-RRB-

-LSB-

-RSB-

R.

G.

Stanton

and

J.

G.

KalbAEeisch

Covering

problems

for

dichotomized

matchings

Aequationes

Math

-LRB-

-RRB-

-LSB-

-RSB-

R.

G.

Stanton

and

J.

G.

KalbAEeisch

Intersection

inequalities

for

the

covering

problem

SIAM

J.

Appl

Math

-LRB-

-RRB-

-LSB-

-RSB-

P.

Tannenbaum

Partitions

of

SIAM

J.

Alg

Discr

Meth

-LRB-

-RRB-

-LSB-

-RSB-

O.

Taussky

and

J.

Todd

Covering

theorems

for

groups

Ann

Soc

Polon

Math

-LRB-

-RRB-

-LSB-

-RSB-

O.

Taussky

and

J.

Todd

Some

discrete

variable

computations

Proc

Sympos

Appl

Math

-LRB-

-RRB-

-LSB-

-RSB-

H.

N.

Ward

A

form

for

J.

Alg

-LRB-

-RRB-

-LSB-

-RSB-

E.

W.

Weber

On

the

football

pool

problem

for

matches

a

new

upper

bound

J.

Combin

Theory

Ser

A

-LRB-

-RRB-

-LSB-

-RSB-

G.

J.

M.

van

Wee

Improved

sphere

bounds

on

the

covering

radius

of

codes

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

G.

J.

M.

van

Wee

More

binary

covering

codes

are

normal

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

G.

J.

M.

van

Wee

Bounds

on

packings

and

coverings

by

spheres

in

qary

and

mixed

Hamming

spaces

J.

Combin

Theory

Ser

A

-LRB-

-RRB-

-LSB-

-RSB-

G.

J.

M.

van

Wee

On

the

non-existence

of

certain

perfect

mixed

codes

Discrete

Math

-LRB-

-RRB-

-LSB-

-RSB-

G.

J.

M.

van

Wee

Covering

Codes

Perfect

Codes

and

Codes

from

Algebraic

Curves

Ph.D.

thesis

Eindhoven

University

of

Technology

The

Netherlands

June

-LSB-

-RSB-

G.

J.

M.

van

Wee

G.

D.

Cohen

and

S.

N.

Litsyn

A

note

on

perfect

multiple

coverings

of

the

Hamming

space

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

G.

J.

M.

van

Wee

Some

new

lower

bounds

for

binary

and

ternary

covering

codes

IEEE

Trans

Inform

Theory

to

appear

-LSB-

-RSB-

L.

T.

Wille

The

football

pool

problem

for

matches

a

new

upper

bound

obtained

by

simulated

annealing

J.

Combin

Theory

Ser

A

-LRB-

-RRB-

-LSB-

-RSB-

S.

K.

Zaremba

A

covering

theorem

for

Abelian

groups

J.

London

Math

Soc

-LRB-

-RRB-

-LSB-

-RSB-

S.

K.

Zaremba

Covering

problems

concerning

Abelian

groups

J.

London

Math

Soc

-LRB-

-RRB-

-LSB-

-RSB-

Z.

Zhang

Linear

inequalities

for

covering

codes

Part

I?pair

covering

inequalities

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

Z.

Zhang

and

C.

Lo

Lower

bounds

on

t

-LSB-

n

k

-RSB-

from

linear

inequalities

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

-LSB-

-RSB-

Z.

Zhang

and

C.

Lo

Linear

inequalities

for

covering

codes

Part

II

triple

covering

inequalities

IEEE

Trans

Inform

Theory

-LRB-

-RRB-

Errata

Paper

-LSB-

-RSB-

Appendix

A

The

-LRB-

-RRB-

code

should

read

Paper

-LSB-

-RSB-

Theorem

The

case

-LRB-

-RRB-

must

be

included

in

the

proof

Paper

-LSB-

-RSB-

Theorem

Should

read

linear

perfect

covering

Theorem

The

-LRB-

-RRB-

code

in

Table

III

is

not

strongly

seminormal

and

Theorem

should

be

deleted

However

the

partitioned

code

has

the

strong

property

that

for

any

x

F

F

P

i

d

-LRB-

x

Ci

-RRB-

and

Theorem

has

been

checked

to

hold

for

the

cases

in

Table

II

where

it

is

used

Theorem

Page

line

should

read

x

F

combinatorial optimization
covering code
covering radius
football pool problem
simulated annealing