数理解析研究所講究録 1228 巻 2001 年 28-34

28Higher Weights for Codes over Finite Rings
Hiroshi Horimoto (堀本 博)
andKeisuke Shiromoto (城本 啓介) *Department of Mathematics, Kumamoto University(熊本大学理学部数理科学科)

1

Introduction
For an $[n, k]$ codeweight (GHW) $d_{r}(C)$over afinite field and $1\leq r\leq k$ , theof is defined by Wei ([10]) as follows:$C$$\mathrm{F}_{q}$$r\mathrm{t}\mathrm{h}$generalized Hamming$C$$d_{r}(C):= \min${$|\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(D)|$:$D$

is a $[n,$$r]$subcode of $C$ },where Supp(D)
and
for
.
Alot of papers dealing with GHW for codes over finite fields have been published (see[9] etc.).
On the other hand, in the last few years, linear codes over finite rings have been in thefocus of the coding research (see [3], [5], [6], [7] and [11], etc.). In particular, Ashikhmin,
Yang, Helleseth et al. ([1], [12], [13] and [4]) introduced thegeneralized Hamming
weight with respect to order (GHWO) $d_{r}(C)$ for alinear code of length overand
$1\leq r\leq\log_{4}|C|$ as follows:$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x):=\{i|x:\neq 0\}$$:=\mathrm{U}\mathrm{x}\mathrm{e}\mathrm{D}\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(\mathrm{x})$$x=(x_{1}, \ldots, x_{n})\in \mathrm{F}_{q}^{n}$$r\mathrm{t}\mathrm{h}$$C$$t(C):= \min$ {$|\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(D)|$

:

$D$

$n$

$\mathbb{Z}_{4}$is asubmodule of $C$ with $\log_{4}|D|=r$ }.

And they exactly determined $d_{r}(C)$ of Preparata, Kerdock, Goethals codes et al. overZ4 for some .
In this paper, we shall introduce aconcept of rank for linear codes over finite chain
rings and consider some fundamental properties of ageneralized Hamming weight withrespect to rank for these codes.
In this paper, all rings are assumed to be finite and associative with $1\neq 0$ . In any
module, 1is assumed to act as the identity.
$r$

Research Fellow of the Japan Society for the Promotion of Science29
2Codes over Finite Chain Rings.
Afinite ring with Jacobson radical $J(R)\neq 0$ is call as achain ring if the principal
left ideals of form achain (see [8] and [5]). We remark that afinite chain ring can. For example, thebe viewed as alocal ring with $J(R)=R\theta$ for anyof integers module , where is aprime power, the Galois ring $GR(q, m)$ ofring
are chain rings. On the otherelements andcharacteristic withare not chain rings. Let, where is not aprime power, andhand,
be the group of units of . In addition,
be the index of nilpotency of $J(R)$ and let
is alocal ring, we denote by aprime power the cardinality of the finite fieldsince$R/J(R)$ , that is,
be the free -module of rank
and $|R|=q^{m}$ . Let
consisting of all -tuples of elements of . With respect to component-wise addition
has the structure of an $(R, R)$ -bimodule. Aright (resp. ,multiplication,and. If isof length over is aright (resp., left) -submodule ofleft) linear code
, then we shall call $C$ as afree code. For aright (left) linear
afree -submodule ofcode over , we define the rank of , denoted by rank(C), as the minimum numberof generators of and define the free rank of $C$ , denoted by -rank(C), as the maximum
rank of the free -submodules of . In this case, is isomorphic, as an module to a
direct sum:
$R$

$R$

$R$

$\theta\in J(R)\backslash J(R)^{2}$$\mathbb{Z}/q\mathbb{Z}$$q$$\mathrm{g}$$\mathrm{F}_{2}+u\mathrm{F}_{2}(u^{2}=0)$$q^{m}$$q$$k$$\mathbb{Z}/k\mathbb{Z}$$\mathrm{F}_{2}+v\mathrm{F}_{2}(v^{2}=v)$$R$$R^{*}$

$m$

$R$

$q$

$R$

$n$

$n$

$R^{n}$$\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}/\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}$

$C$

$R$

$C$

$R$

$R^{n}$$R/J(R)\cong \mathrm{F}_{q}$

$R$

$R$

$n$

$R^{n}$$C$$R^{n}$

$C$

$R$

$C$

$\mathrm{f}$

$C$

$R$

$R$

$\mathrm{C}$$C\cong\oplus(R/R\theta^{i})^{k_{i}}i=1m$,

where $R\theta^{i}:=\{r\theta^{i}|r\in R\}=\{x\in R|x\theta^{m-i}=0\}$ , for each $i\in\{1,2, \ldots, m\}$ . We
and f-rank(C) $=k_{m}$ , and define the type of $C$ , denoted bynote that rank(C). For an $R$ module $M$ , the socte of $M$ , that is,
type (C), as the sequence
the sum of all simple submodules of $M$ , is denoted by Soc(M). For aright (resp., left)
linear code $C$ over $R$ , we note that$=\Sigma_{i=1}^{m}k_{i}$$(k_{1}, k_{2}, \ldots, k_{m})$Soc(C) $=\{x\in C|x\theta=0\}$(resp., Soc $(\mathrm{C}17)=\{x\in C|\theta x=0\}$ ).
For aright (left) linear code over $R$ , we define $I(C)$ as aminimal free R-submodule
which contains $C$ and define $F(C)$ as amaximal free $R$-submodule of $C$ . If $C$ isofaright (resp., left) linear code of length over $R$ , then $I(C)$ is aright (resp., left) free
and $F(C)$ is aright (resp., left) free code ofcode of length with rank(/(C))-rank(C)(cf. [7]).
length with rank(F(C))
For avector $x=$ $(x_{1}, \ldots, x_{n})\in R^{n}$ , the support of is defined by$C$$R^{n}$

$n$

$n$

$n$

$=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(C)$$=\mathrm{f}$$x$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x):=\{i|x_{i}\neq 0\}$

30
and the Hamming weightof r is defined to be the order of the support of z. The
minimum Hamming weight of alinear code C of length n over R is$\mathrm{w}\mathrm{t}(\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT})$$d(C):= \min\{\mathrm{w}\mathrm{t}(x)|(0\neq)x\in C\}$.

If Soc(R)?) $\cong R/J(R)$ as right -modules and as left -modules, then is called as a
Frobenius ring ([8], [7] and [11]). Since achain ring is aFrobenius ring, we have an Risomorphism :Soc(ff) $\cong R/J(R)$ . In this case, induces the following R-isomorphism:
$R$

$R$

$R$

$R$

$\phi$$\phi$$\phi^{n}$: Soc $(R)^{n}\cong(R/J(R))^{n}$:$x=(x_{1}, \ldots, x_{n})\mapsto\phi^{n}(x)=(\phi(x_{1}), \ldots, \phi(x_{n}))$,(cf. [8] and [7]). We have the following proposition.

Proposition 2.1 ([7]) If C is a right (left) linear code of length n over R, thenis a linear [n, rank(C), $d(C)$ ] code over the finite field $R/J(R)$ .
$\phi^{n}(\mathrm{S}\mathrm{o}\mathrm{c}(C))$

For two vectors $x=$
product$(x_{1}, \ldots, x_{n})\in R^{n}$

and $y=(y_{1}, \ldots, y_{n})\in R^{n}$ , we define the inner$\langle x, y\rangle:=x_{1}y_{n}+\cdots+x_{n}y_{n}$For asubset
follows:$C\subseteq R^{n}$, we define the right dual code

$C^{[perp]}$$:=$$[perp]_{C}:=$

If

$C$

{

$y\in R^{n}|\langle x$$C^{[perp]}$.

and the left dual $codeC[perp]$ ofas, $y\rangle=0$ , Vt $\in C$ }$\{y\in R^{n}|\langle y, x\rangle=0, \forall x\in C\}$is aright (resp., left) linear code of length
rank

$C$

$n$

over

$R$

.

, then-rank $(^{[perp]}C)=n$$(C)+\mathrm{f}$(resp., rank(C)-rank $(C^{[perp]})=n$ )$+\mathrm{f}$and(resp., $[perp](C^{[perp]})=C$ ) (cf. [5] and [7]).
Agenerator matrix of aright (resp., left) linear code $C$ of length
over
is a
rank(C)
matrix over whose rows form aminimal set of generators of . Similarly,
aparity check matrix of is an(-rank(C))matrix over whose columns form
aminimal set of generators of(resp.,).
In the remaining part of this paper, we shall concentrate on right linear codes because
all results and proofs for left linear codes always go through as well as those for rightlinear code$(^{[perp]}C)^{[perp]}=C$

$R$

$n$

$\cross n$

$R$

$C$

$C$

$n\cross$$[perp] C$$n-\mathrm{f}$$C^{[perp]}$

$R$

31
3Generalized Hamming Weights
For asubset$C\subseteq R^{n}$, we define the support of $C$ bySupp(C)

Evidently we note that if$|\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(C_{1})|\leq|\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(C_{2})|$$C_{1}$

and$C_{2}$$:= \bigcup_{X\in C}$

Supp (C)are subsets ofsuch that$R^{n}$$C_{1}\subseteq C_{2}$.

Definition 3.1 For aright linear code of length over andgeneralized Hamming weight with respect to rank (GHWR) of
$C$

$R$

$n$

$r\mathrm{t}\mathrm{h}$$dr\{C$ ) $:= \min${, then$|\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(D)|$

:

$D$

is an

-submodule of$R$

The weight hierarchy of $C$ is the set of integers$1\leq r\leq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(C)$

$C$

, the

is defined bywith rank(D) $=r$ }.

$C$$\{d_{r}(C) : 1\leq r\leq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(C)\}$The following lemma is essential.
Lemma 3.2 If$C$is a right (left) linear code of length$)$

Soc(C)$=\mathrm{S}\mathrm{o}\mathrm{c}(I(C))$

$n$

over

$R$

, then.

The following result is ageneralization of Proposition 2.1 with respect to GHWR.

Theorem 3.3 Let $C$ be a right linear code$C$of length$d_{r}(C)=d_{r}(\mathrm{S}\mathrm{o}\mathrm{c}(C))=d_{r}(I(C))$

for any ,$r$$1\leq r\leq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(C)$over R. Then

$n$

,

.

Remark 3.4 The above theorem also claims that all free -submodules ofcontain C and have the same rank as C have the same weight hierarchy as C.
$R$$R^{n}$

which

Using the above theorem, we have the following results from Theorem 1and Corollary
1in [10].
Corollary 3.5 For a right linear code $C$ of length

$n$

over

with rank(C) $=k>0$ ,$R$$1\leq d_{1}(C)<d_{2}(C)<\cdots<d_{k}(C)\leq n$

$\}$

.

Corollary 3.6 For a right linear code C of length n over R and any r,$d_{r}(C)\leq n$-rank(C)+r.

$1\leq r\leq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(C)$,32
n –rank(C)+r, then C is called an rth
If C meets the above bound, i.e.,
MDS code over R. In [2] and [7], the first MDS codes over the finite rings (simply called
MDR or MDS codes in these papers) are studied.
For aright linear code C of length n and M N{1,2,72}, we set$d_{r}(C)\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$\ldots ,$R^{n}(M)$$:=$$C(M)$$\{x\in R^{n}|\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)\subseteq M\}$$:=C\cap R^{n}(M)=\{x\in C|\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(x)\subseteq M\}$.

Clearly, $R^{n}(M)$ is afree -module of rank $|M|$ and $C(M)$ is also aright linear code oflength over . And for right linear codes and $D$ over and alinear map : $Carrow D$ ,we define
$R$

$n$

$C$

$R$

C’
$\psi^{*}$

$:=$

$R$

$\psi$$\mathrm{H}\mathrm{o}\mathrm{m}_{R}(C, R)$:$D^{*}arrow C^{*}$:$g\mapsto g\psi$.

Moreover, there is the following isomorphism as left i2-m0du1es:$f$:$R^{n}arrow(R^{n})^{*}$:$x\mapsto(f(x) :y\mapsto\langle x, y\rangle)$.

Then the following proposition is essential.
Proposition 3.7 ([6]) Let$C$

be a right linear code of length$n$over R. Then the sequence$0arrow[perp] C(N-M)arrow R^{n}(N-M)\mathrm{i}\mathrm{n}\mathrm{c}arrow C^{*}arrow C(M)^{*}f\mathrm{r}\mathrm{a}\mathrm{e}arrow 0$

is exact as left $R$ -modules for any $M\subseteq N$ , where the maps inc;map, the restriction map, respectively.

$\mathrm{r}\mathrm{e}\mathrm{s}$denote the inclusion

In [6], they proved the Singleton type bound for codes over finite quasi-Frobenius
rings by using this proposition. In this paper, we prove aduality for GHWR of codes
over finite chain rings using this proposition.

Lemma 3.8 Let$C$

be a right linear code of length{$d_{r}(C)= \min$ $|M|$

for all ,$r$$1\leq r\leq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(C)$$n$over R. Then

: rank(C(M)) $=r$ ,$M\subseteq N$},.

Aduality for GHW of codes over finite fields is proved in [10] and similarly, aduality
for GHWO of codes over Galois rings is proved in [1]. As in these case, we have asimilar
duality relation for GHWR of codes over finite chain rings as follows

33
Theorem 3.9 Let $C$ be a right linear code of length

over

$n$

$R$

with rank(C) $=k$ . Then$\{d_{r}(C) : 1\leq r\leq k\}=\{1,2, \ldots, n\}\backslash \{n+1-d_{r}(F(^{[perp]}C)) : 1\leq r\leq n-k\}$Though we have many possibilities of taking $F(C)$ for aright linear code
following result follows from the above theorem.

.
$C$

, the

Corollary 3.10 If $C$ is a right linear code of length over $R$ with f-rank(C) $=k_{0}$ , then
all right free -submodules of $C$ with rank
have the same weight hierarchy determinedby that.
$n$$R$$k_{0}$$of^{[perp]}C$

Now we introduce aweight for avector inwhich is ageneralization of the Lee$(0\neq)x\in
R$ , we define the socle weight $s(x)$ ofweight for avector in . For an element$(0\neq)x\in R$ as follows:$R^{n}$$\mathbb{Z}_{4}^{n}$$s(x)=\{$$q-1$$(x\not\in \mathrm{S}\mathrm{o}\mathrm{c}(R))$’$q$$(x\in \mathrm{S}\mathrm{o}\mathrm{c}(7?))$and set $s(0)=0$ . For example, if $R=\mathbb{Z}_{27}=\{0,1,2, \ldots, 26\}$ , then $s(x)=2$ for $x\neq 0,9,18$
and $s(x)=3$ for $x=9,18$ . For avector $x=(x_{1}, \ldots, x_{n})\in R^{n}$ , the socle weight $w_{S}(x)$ ofis defined by$x$$w_{S}(x):= \sum_{j=1}^{n}s(x_{j})$For aright linear code $C$ of length
defined as follows:

$n$

over

$R$

.

, the minimum socle weight $d_{S}(C)$ of $C$ is$d_{S}(C):= \min\{w_{S}(x)|(0\neq)x\in C\}$ .

Lemma 3.11 Let
be a right linear code of length and of rank overand letarray of all codewords in C. Then each column ofbe thecorresponds
to the following case: the column contains all elements ofequally often for any
$C$

$A$

$R$

$k$

$n$

$A$

$|C|\mathrm{x}n$$R\theta^{i}$$i\in\{0,1, \ldots, m-1\}$ .

Then we have the following theorem. Asimilar result for Lee weights of codes overcan be found in [12] and corresponds to the special casein the following result.

$\mathbb{Z}_{4}$$R=\mathbb{Z}_{4}$Theorem 3.12 Let $C$ be a right linear code of length$n$over R. Then we have$| \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(C)|=\frac{1}{|C|(q-1)}\sum_{x\in C}w_{S}(x)$Corollary 3.13 If
$C$ ,

$C$

$1\leq r\leq \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(\mathrm{C})_{;}$

be a right linear code of length$n$satisfies$d_{r}(C) \geq\lceil\frac{(q^{r}-1)d_{S}(C)}{q^{r}(q-1)}\rceil$where$\lceil a\rceil$.

over $R$ , then the $rth$ GHWR of,denotes the smallest integer greater than or equal to .
$a$34
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[7] H. Horimoto and K. Shiromoto, MDS codes over finite quasi-Probenius rings, submitted.
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[10] V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inforrm.
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