binary 18 11 2 codes do not exist nor do 64 53 2 codes a binary linear block code c with block length n and dimension n is commonly denoted by n k or if its minimum distance is d by n k d the code s covering radius r c can be defined as the smallest number r such that any binary column vector of length n k can be written as a sum of r or fewer columns of a parity check matrix of c an n k code with covering radius r is denoted by n k r r a brualdi et al 1989 showed that l m r is defined to be the smallest n such that an n n m r code exists l m 2 is known for m or 6 while it is shown by brualdi et al that 17 or l

a lower bound on the undetected error probability and strictly optimal codes error detection is a simple technique used in various communication and memory systems to enhance reliability we study the probability that a q ary linear or nonlinear block code of length n and size m fails to detect an error a lower bound on this undetected error probability is derived in terms of q n and m the new bound improves upon other bounds mentioned in the literature even those that hold only for linear codes block codes whose undetected error probability equals the new lower bound are investigated we call these codes strictly optimal codes and give a combinatorial characterization of them we also present necessary and sufficient conditions for their existence in particular we find all values of n and m for which strictly optimal binary codes exist and determine the structure of all of them for example we construct strictly optimal binary coded decimal codes of length four and five and we show that these are the only possible lengths of such codes

a 2 adic approach to the analysis of cyclic codes this paper describes how 2 adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over z sub 2 a a spl ges 2 the ring of integers modulo 2 sup a it provides a 2 adic proof of a theorem of mceliece that characterizes the possible hamming weights that can appear in a binary cyclic code a generalization of this theorem is derived that applies to cyclic codes over z sub 2 a that are obtained from binary cyclic codes by a sequence of hensel lifts this generalization characterizes the number of times a residue modulo 2 sup a appears as a component of an arbitrary codeword in the cyclic code the limit of the sequence of hensel lifts is a universal code defined over the 2 adic integers this code was first introduced by calderbank and sloane 1995 and is the main subject of this paper binary cyclic codes and cyclic codes over z sub 2 a are obtained from these universal codes by reduction modulo some power of 2 a special case of particular interest is cyclic codes over z sub 4 that are obtained from binary cyclic codes by means of a single hensel lift the binary images of such codes under the gray isometry include the kerdock preparata and delsart goethals codes these are nonlinear binary codes that contain more codewords than any linear code presently known fundamental understanding of the composition of codewords in cyclic codes over z sub 4 is central to the search for more families of optimal codes this paper also constructs even unimodular lattices from the hensel lift of extended binary cyclic codes that are self dual with all hamming weights divisible by 4 the leech lattice arises in this way as do extremal lattices in dimensions 32 through 48

convergence in mean for els based adaptive control abstract almost sure global convergence results are available for stochastic adaptive control this note looks at the nontrivial technical details required to ensure corresponding convergence in p th mean results both convergence results are strong but are not implied by the other and so together almost sure and p th mean convergence are stronger than either the other aim of this paper is to show that the combination of these two forms of convergence results makes the stability results more robust and potentially of more practical significance

on binary cyclic codes of odd lengths from 101 to 127 all binary cyclic codes of odd lengths are checked from 101 to 127 to find codes which are better than those in a table by t verhoeff 1989 there are five such cases namely 117 36 32 117 37 29 117 42 26 117 49 24 and 127 36 35 cyclic codes according to verhoeff s table the previously known ranges of the highest minimum distance were 28 40 28 40 25 37 22 32 and 32 46 respectively applying constructions x and y1 120 37 32 and 108 28 32 codes were found moreover the highest minimum distances that cyclic codes of length 127 can attain are determined

using combinatorial optimization to design good unit memory convolutional codes a method for designing good unit memory convolutional codes is presented the method is based on the decomposition of the original problem into two easier subproblems that can be formulated as optimization problems and solved by efficient heuristic search algorithms the efficacy of this method is demonstrated by a table containing 33 new unit memory convolutional codes n k with 5

tailbiting codes bounds and search results tailbiting trellis representations of linear block codes with an arbitrary sectionalization of the time axis are studied the notations of regular and irregular tailbiting codes are introduced and their maximal state complexities are lower bounded the asymptotic behavior of the derived bound is investigated furthermore for regular tailbiting codes the product state complexity is lower bounded tables of new tailbiting trellis representations of linear block codes of rates 1 2 1 3 and 1 4 are presented almost all found trellises are optimal in the sense of the new bound on the state complexity and for most codes with nonoptimal trellises there exist time varying trellises which are optimal five of our newly found tailbiting codes are better than the previously known linear codes with the same parameters four of them are also superior to any previously known nonlinear code with the same parameters also more than 40 other quasi cyclic codes have been found that improve the parameter set of previously known quasi cyclic codes

adaptation of the mactaggart and jack complex multiplication algorithm for floating point operators with a suitable treatment of the exponents in the input operands a hardware implementation of the mactaggart and jack fixed point complex multiplication algorithm can also calculate a floating point product with no loss in accuracy from the greater dynamic range of the floating point inputs this floating point technique can be extended to any sum to two products operation such as encountered in matrix multiplication and vector cross products

the trace operator and redundancy of goppa codes the dimension of subfield subcodes and in particular generalized goppa codes is studied using the kernel of an associated trace map bounds on the dimension for a particular general class of binary goppa codes are obtained deeper study of some special cases and other related polynomials yields still tighter bounds for any s 1

nine good rate m 1 pm quasi cyclic codes the class of quasi cyclic qc codes has been proven to contain many good codes to date the known results are largely codes of the form 1 p and p 1 p constructed from circulant matrices a generalization of these rate 1 p codes to rate m 1 pm codes based on the theory of 1 generator qc codes is presented the results of a search for good codes based on heuristic combinatorial optimization are nine codes which improve the known lower bounds on the minimum distance of binary linear codes

generalized hamming weights of nonlinear codes and the relation to the z sub 4 linear representation we give a new definition of generalized hamming weights of nonlinear codes and a new interpretation connected with it these generalized weights are determined by the entropy length profile of the code we show that this definition characterizes the performance of nonlinear codes on the wire tap channel of type ii the new definition is invariant under translates of the code it satisfies the property of strict monotonicity and the generalized singleton bound we check the relations between the generalized weight hierarchies of z sub 4 linear codes and their binary image under the gray map we also show that the binary image of a z sub 4 linear code is a symmetric not necessarily rectangular code moreover if this binary image is a linear code then it admits a twisted squaring construction

a table of state complexity bounds for binary linear codes this article contains a table of bounds on the state complexity of binary linear codes with length smaller than 25 general results on the state complexity of binary linear codes with low dimension or low minimum distance are included

new binary linear block codes corresp using some known techniques several new binary linear codes are constructed i e codes with a greater minimum distance than any previously known binary code of the same length and dimension 8 a detailed description of one of the constructions is given

generalized hamming weights of linear codes the generalized hamming weight d sub r c of a binary linear code c is the size of the smallest support of any r dimensional subcode of c the parameter d sub r c determines the code s performance on the wire tap channel of type ii bounds on d sub r c and in some cases exact expressions are derived in particular a generalized griesmer bound for d sub r c is presented and examples are given of codes meeting this bound with equality

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

on the 162 8 80 codes constructions of 162 8 80 and 159 8 78 codes are given this solves the open problems of finding the minimum length of binary codes of dimension s and minimum distances 78 and 80 respectively

quasi cyclic unit memory convolutional codes unit memory convolutional codes with generator matrices which are composed of circulant submatrices are introduced this structure facilitates the analysis of efficient search for good codes equivalences among such codes and some of the basic structural properties are discussed in particular catastrophic encoders and minimal encoders are characterized and dual codes treated further various distance measures are discussed and a number of good codes some of which result from efficient computer search and some of which result from known block codes are presented

two new rate 2 p binary quasi cyclic codes a class of rate 2 p quasi cyclic codes can be characterized in terms of m spl times m circulant matrices in this correspondence two new codes with parameters 80 10 35 and 95 10 42 are presented which improve the known lower bound on the maximum possible minimum distance the former code can be extended with an even parity check bit to an 81 10 36 code that establishes that d sub 2 81 10 36

convergence analysis of an extended krylov subspace method for the approximation of operator functions in exponential integrators we analyze the convergence of an extended krylov subspace method for the approximation of operator functions that appear in exponential integrators for operators the size of the polynomial part of the extended krylov subspace is restricted according to the smoothness of the initial data this restriction for the continuous operator has a significant influence on the approximation of matrix functions evaluated for matrices stemming from space discretizations of the continuous operator we prove convergence of the method for the continuous operator and in the discrete case this leads to a convergence independent of the norm of the discretized operator uniformly over all possible grids the analysis is illustrated by numerical experiments

optimal and near optimal encoders for short and moderate length tail biting trellises the results of an extensive search for short and moderate length polynomial convolutional encoders for time invariant tail biting representations of block codes at rates r 1 4 1 3 1 2 and 2 3 are reported the tail biting representations found are typically as good as the best known block codes

a sequential decoder for linear block codes with a variable bias term metric a sequential decoder for linear block codes that performs maximum likelihood soft decision decoding is described the decoder uses a metric computed from a lower bound on the cost of the unexplored portion of the code tree it is shown that for certain block codes the average computational complexity of this metric is superior to that of the fano metric a new function the cumulative column distance function is introduced for linear block codes this function is an important factor that determines the average computational effort of a sequential decoder for a linear block code with an arbitrary maximum likelihood metric simulation results show that a sequential decoder for linear block codes with a fast growing cumulative column distance function achieves a low computational complexity a result analogous to that for convolutional codes

the trace description of irreducible quasi cyclic codes the notion of a q ary irreducible quasi cyclic code of block length n and index r is introduced a trace description of such a code is provided in a fashion similar to the trace description of irreducible cyclic codes in particular it is shown that an irreducible quasi cyclic code of dimension k is completely described by an irreducible cyclic code and r elements from a field of cardinality q sup k using this fact a number of binary irreducible quasi cyclic codes of index 2 are constructed and their weight spectra obtained

on the structure of linear codes with covering radius two and three we obtain new bounds on l m r the minimum length of a linear code with codimension m and covering radius r all bounds are derived in a uniform way we employ results from coding theory some earlier results on covering codes and combinatorial arguments we prove the following bounds l 6 2 13 l 7 2 19 l 8 2 spl ges 25 l 9 2 spl ges 34 l 2m l 2 spl ges 2 sup m 1 for m spl ges 3 l 14 2 spl ges 182 l 16 2 spl ges 363 l 18 2 spl ges 725 l 20 2 spl ges 1449 l 22 2 spl ges 2897 l 24 2 spl ges 5794 l 9 3 spl ges 17 l 10 3 spl ges 21 l 12 3 spl ges 31 l 13 3 spl ges 38

the preparata and goethals codes trellis complexity and twisted squaring constructions the trellis complexity of the preparata and goethals codes is examined it is shown that at least for a given set of permutations these codes are rectangular upper bounds on the state complexity profiles of the preparata and goethals codes are given the upper bounds on the state complexity of the preparata and goethals codes are determined by the dimension length profiles dlp of the extended primitive double and triple error correcting bch codes respectively a twisted squaring construction for the preparata and goethals codes is given based on the double and triple error correcting extended primitive bch codes respectively

two new nonlinear binary codes we present two new binary codes a 25 384 9 code and a 49 393216 13 code these codes are better than the currently known binary codes with the same length and minimum distance

constructions of generalized concatenated codes and their trellis based decoding complexity in this article constructions of generalized concatenated gc codes with good rates and distances are presented some of the proposed gc codes have simpler trellis complexity than euclidean geometry eg reed muller rm or bose chaudhuri hocquenghem bch codes of approximately the same rates and minimum distances and in addition can be decoded with trellis based multistage decoding up to their minimum distances several codes of the same length dimension and minimum distance as the best linear codes known are constructed

extremal binary self dual codes in this correspondence we investigate binary extremal self dual codes numerous extremal self dual codes and interesting self dual codes with minimum weight d 14 and 16 are constructed in particular the first extremal type i 86 43 16 code and new extremal self dual codes with weight enumerators which were not previously known to exist for lengths 40 50 52 and 54 are constructed we also determine the possible weight enumerators for extremal type i codes of lengths 66 100

on unit constraint length convolutional codes unit constraint length convolutional codes are codes that can be generated by encoders with a single delay element the maximal free distances of these codes are determined in terms of the maximal minimum distances of linear block codes a formula for the weight enumerator of unit constraint length convolution codes is derived and a macwilliams identity to determine the weight enumerator state diagram of a code from the weight enumerator state diagram of its dual code is given

the nordstrom robinson code representation over gf 4 and efficient decoding we show that the nordstrom robinson 1968 code may be represented as the union of binary images of two isomorphic linear 4 2 3 codes over gf 4 certain symmetries of the unique quaternary 4 2 3 quadracode are discussed it is shown how the properties of the nordstrom robinson code itself such as weight distribution distance invariance and the well known representation as the union of eight cosets of the 16 5 8 reed muller code may be easily rederived from these symmetries in addition we introduce a decoding algorithm for the nordstrom robinson code which involves projecting its codewords onto the codewords of the quadracode the algorithm is simple enough to enable hard decision decoding of the nordstrom robinson code by hand furthermore we present an algorithm for maximum likelihood soft decision decoding of the nordstrom robinson code based on its representation over gf 4 the complexity of the proposed algorithm is at most 205 real operations this is to the best of our knowledge less than the complexity of any existing decoder and twice as efficient as the fast maximum likelihood decoder of adoul 1987 we also investigate several related topics such as bounded distance decoding sphere packings and the 20 2048 6 code

on the generalized hamming weights of cyclic codes we prove several results on the generalized hamming weights ghw s of linear codes particularly for cyclic codes based on these and previously known results we give some efficient algorithms for computing ghw hierarchy of cyclic codes we give complete weight hierarchy for each of the binary cyclic codes of odd lengths spl les 31 a table of second and third ghw s of binary cyclic codes of odd lengths spl les 57 is also presented we have also computed the second ghw of all binary cyclic codes of length 63 and the third ghw of one code from each dimension

a method for extending binary linear codes corresp five binary linear n k d codes are presented that contain more words than the codes with corresponding values of n and d that are listed in the table of macwilliams and sloane the construction method is a generalization of construction x of sioane reddy and chen

binary codes derived from the hoffman singleton and higman sims graphs some binary linear codes of length 50 and 100 are constructed using the adjacency matrices of the hoffman singleton graph and the higman sims graph some of the codes are optimal or nearly optimal for the given length and dimension the dual codes admit majority logic decoding

constructions and bounds for systematic tec aued codes several methods of constructing systematic t error correcting all unidirectional error detecting codes are described these codes can be constructed by adding a tail to a linear t error correcting code but other constructions presented are more of an ad hoc nature these codes will often be found as suitably chosen subsets of nonsystematic tec aued codes further bounds on the word length of systematic tec aued codes are derived and extensive tables are given

bounds on the dimension of codes and subcodes with prescribed contraction index abstract let c be a linear code over gf q spanned by the rows of a matrix g of rank k a nonnegative integer is said to be the contraction index of c if a maximal set of pairwise linearly independent columns of g has k elements we derive several upper and lower bounds on the dimension of a proper subcode of c with a prescribed contraction index v q of length n minimum hamming distance d and contraction index for certain values of n and d the latter bound is shown to be tight for all q and this substantially generalizes the results obtained by delsarte and by duc for 1

a 55 16 19 binary goppa code and related codes having large minimum distance a 55 16 19 binary goppa code is used to construct 57 17 17 58 17 18 59 17 19 and 60 17 20 codes the first two codes have smaller redundancy than previously known codes linear or nonlinear of the same length and minimum distance the last two codes have parameters previously attained only by nonlinear codes

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

tailbiting codes obtained via convolutional codes with large active distance slopes the slope of the active distances is an important parameter when investigating the error correcting capability of convolutional codes and the distance behavior of concatenated convolutional codes the slope of the active distances is equal to the minimum average weight cycle in the state transition diagram of the encoder a general upper bound on the slope depending on the free distance of the convolutional code and new upper bounds on the slope of special classes of binary convolutional codes are derived moreover a search technique resulting in new tables of rate r 1 2 and rate r 1 3 convolutional encoders with high memories and large active distance slopes is presented furthermore we show that convolutional codes with large slopes can be used to obtain new tailbiting block codes with large minimum distances tables of rate r 1 2 and rate r 1 3 tailbiting codes with larger minimum distances than the best previously known quasi cyclic codes are given two new tailbiting codes also have larger minimum distances than the best previously known binary linear block codes with same size and length one of them is also superior in terms of minimum distance to any previously known binary nonlinear block code with the same set of parameters

lower bounds on t n k from linear inequalities the linear inequality method for covering codes is used to improve the lower bounds of t n k the smallest covering radius of any n k binary linear code to make better use of the strength of this method the relation between the covering radius of a code and the minimum distance of its dual code is studied the authors obtained 65 improved lower bounds for t n k within the range of n

new minimum distance bounds for certain binary linear codes let an n k d code denote a binary linear code of length n dimension k and minimum distance at least d define d n k as the maximum value of d for which there exists a binary linear n k d code t verhoeff 1989 has provided an updated table of bounds on d n k for 1

a beast for prowling in trees when searching for convolutional codes and tailbiting codes of high complexity it is of vital importance to use fast algorithms for computing their weight spectra which corresponds to finding low weight paths in their code trellises this can be efficiently done by a combined search in both forward and backward code trees a bidirectional efficient algorithm for searching such code trees beast is presented for large encoder memories it is shown that beast is significantly more efficient than comparable algorithms beast made it possible to find new convolutional and tailbiting codes that have larger free minimum distances than the previously best known codes with the same parameters tables of such codes are presented

inequalities for covering codes any code c with covering radius r must satisfy a set of linear inequalities that involve the lloyd polynomial l sub r x these generalize the sphere bound syndrome graphs associated with a linear code c are introduced to help keep track of low weight vectors in the same coset of c if there are too many such vectors c cannot exist illustrations show that t 17 10 3 and t 23 15 3 where t n k is the smallest covering radius of any n k code

two new binary codes obtained by shortening a generalized concatenated code the authors construct a 75 13 30 code and a 75 11 32 code the minimum distances of these codes are one larger than the largest known t verhoeff 1987 minimum distances of codes with the same length and dimension as a 75 11 33 code does not exist the 75 11 32 code is in a sense optimal both codes are obtained by judiciously shortening a 80 14 32 code that is obtained as a generalized concatenated code

linear codes in generalized construction of resilient functions with very high nonlinearity we provide a new generalized construction method for highly nonlinear t resilient functions f f sub 2 sup n spl rarr f sub 2 sup m the construction is based on the use of linear error correcting codes together with highly nonlinear multiple output functions given a linear u m t 1 code we show that it is possible to construct n variable m output t resilient functions with very high nonlinearity for n u the method provides the currently best known nonlinearity results for most of the cases

a note on generalized hamming weights of bch 2 determines results for the first six generalized hamming weights of double error correcting primitive binary bch codes

on the bcjr trellis for linear block codes in this semi tutorial paper we will investigate the computational complexity of an abstract version of the viterbi algorithm on a trellis and show that if the trellis has e edges the complexity of the viterbi algorithm is spl theta e this result suggests that the best trellis representation for a given linear block code is the one with the fewest edges we will then show that among all trellises that represent a given code the original trellis introduced by bahl cocke jelinek and raviv in 1974 and later rediscovered by wolf 1978 massey 1978 and forney 1988 uniquely minimizes the edge count as well as several other figures of merit following forney and kschischang and sorokine 1995 we will also discuss trellis oriented or minimal span generator matrices which facilitate the calculation of the size of the bcjr trellis as well as the actual construction of it

lower bounds on trellis complexity of block codes the trellis state complexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis we present a new lower bound on the state complexity of linear codes which includes most of the existing bounds as special cases the new bound is obtained by dividing the time axis for the code into several sections of varying lengths as opposed to the division into two sections the past and the future employed in the well known dlp bounds for a large number of codes this results in a considerable improvement upon the dlp bound moreover we generalize the new bound to nonlinear codes and introduce several alternative techniques for lower bounding the trellis complexity based on the distance spectrum and other combinatorial properties of the code we also show how the general ideas developed in this paper may be employed to lower bound the maximum and the total number of branches in the trellis leading to considerably tighter bounds on these quantities furthermore the asymptotic behavior of the new bounds is investigated and shown to improve upon the previously known asymptotic estimates of trellis state complexity

new bounds on binary linear codes of dimension eight corresp let n k d be the smallest integer n such that a binary linear code of length n dimension k and minimum distance at least d exists new results are given that improve the best previously known bounds on n 8 d

the nonexistence of certain binary linear codes some of the upper bounds given by t verhoeff 1987 are improved by proving the nonexistence of codes with certain parameters necessary preliminary results are stated the results give rise to upper bounds on d n k for many further values of n k by using simple standard techniques the further bounds would automatically follow from theorems by using verhoeff s computerized updating program but the authors state them here because several of them are required in the proofs of subsequent theorems

new binary codes constructed by an old technique corresp applying a known concatenation procedure to a 30 15 12 quadratic residue code over gf 4 results in 29 new binary codes each of these codes is larger than previous codes having the same length and minimum distance

trellis decoding complexity of linear block codes in this partially tutorial paper we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity maximum state and edge dimensions total span length and total vertices edges and mergers we obtain bounds on these complexities as extensions of well known dimension length profile dlp bounds codes meeting these bounds minimize all the complexity measures simultaneously conversely a code attaining the bound for total span length vertices or edges must likewise attain it for all the others we define a notion of uniform optimality that embraces different domains of optimization such as different permutations of a code or different codes with the same parameters and we give examples of uniformly optimal codes and permutations we also give some conditions that identify certain cases when no code or permutation can meet the bounds in addition to dlp based bounds we derive new inequalities relating one complexity measure to another which can be used in conjunction with known bounds on one measure to imply bounds on the others as an application we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths

binary even 25 15 6 codes do not exist corresp that binary even 25 15 6 codes do not exist is proved by passing from coding theory to bf f 2 affine geometry all 25 sets in bf f 9 2 are shown to contain affine planes

trellis oriented decomposition and trellis complexity of composite length cyclic codes the trellis complexity of composite length cyclic codes clcc s is addressed we first investigate the trellis properties of concatenated and product codes in general known factoring of clcc s into concatenated subcodes is thereby employed to derive upper bounds on the minimal trellis size and state space profile new decomposition of clcc s into product subcodes is established and utilized to derive further upper hounds on the trellis parameters the coordinate permutations that correspond to these bounds are exhibited additionally new results on the generalized hamming weights of clcc s are obtained the reduction in trellis complexity of many clcc s leads to soft decision decoders with relatively low complexity

sphere packing bounds for convolutional codes we introduce general sphere packing bounds for convolutional codes these improve upon the heller 1968 bound for high rate convolutional codes for example based on the heller bound mceliece 1998 suggested that for a rate n 1 n convolutional code of free distance 5 with spl nu memory elements in its minimal encoder it holds that n spl les 2 sup spl nu 1 2 a simple corollary of our bounds shows that in this case n 2 sup spl nu 2 an improvement by a factor of spl radic 2 the bound can be further strengthened note that the resulting bounds are also highly useful for codes of limited bit oriented trellis complexity moreover the results can be used in a constructive way in the sense that they can be used to facilitate efficient computer search for codes

on generator matrices of codes the if class of the q ary linear codes of given length dimension and minimum weight is nonempty it is shown to contain a code whose generator matrix consists of words of minimum weight

optimal binary linear codes and z sub 4 linearity we give necessary and sufficient conditions for a binary linear code to be z sub 4 linear especially we treat optimal binary linear codes and determine all such codes with minimum weight less or equal to six which are z sub 4 linear

new linear codes constructed by concatenating extending and shortening methods corresp blokh and zyablov s concatenating method constructions x y1 of sloane et al and aiitop s construction xx are applied to various codes to obtain new binary linear codes these codes improve on the recent updated helgert stinaff table of verhoeff

a 55 16 19 binary goppa code corresp a 55 16 19 binary goppa code and some related codes are described these codes are found to contain more codewords than any previously known codes of the same length and minimum distance

some new binary quasi cyclic codes by means of local search techniques five quasi cyclic codes have been found that have a higher minimum distance than known binary linear codes the new codes have parameters 102 17 37 60 20 17 84 21 27 105 21 36 and 100 25 30 also 39 other quasi cyclic codes have been found that improve the parameter sets of previously known quasi cyclic codes twenty four of them give a new and easier description of binary linear codes with best known parameters

new optimal binary linear codes of dimensions 9 and 10 eighteen new codes are presented which improve the bounds on maximum minimum distance for binary linear codes they are rate m r pm r spl ges 1 degenerate quasi cyclic qc codes based on the known upper bounds six of these new codes are optimal in addition five two weight qc codes of dimension 8 are given

some best rate 1 p and rate p 1 p systematic quasi cyclic codes tables of 1 p and rate p 1 p binary quasi cyclic qc codes that extend previously published results are presented many of these codes attain bounds given by t verhoeff 1987 who composed a table of bounds on the maximum possible minimum distance of binary linear codes many of the codes presented meet or improve these bounds a best code is considered to be one which has the largest possible minimum distance for the given code dimensions n and k and class of error correcting codes a good code has the maximum known minimum distance for the class of codes an optimal code is one that achieves the maximum possible minimum distance for a linear code with the same dimensions binary power residue codes are found and used to construct qc codes

the concatenated structure of cyclic and abelian codes it is shown that a linear two dimensional cyclic code of length nn can be factorized into a direct sum of concatenated codes with cyclic inner and outer codes and conversely thai a two dimensional cyclic code can be constructed in this way this result is extended and it appears that the abelian codes are obtainable by taking a direct sum of several concatenations of cyclic codes codes are constructed that are better than any previously known in particular low rate cyclic codes superior to the duals of high rate bch codes are constructed

codes satisfying the chain condition the authors considered weight hierarchies of codes satisfying the chain condition they called these chain good first they gave a set of simple necessary conditions for a sequence to be chain good they proved that given one chain good sequence there is an infinite set of chain good sequences that can be constructed from this one sequence finally they used this result to completely describe the sets of chain good sequences of dimensions up to 5

the smallest length of eight dimensional binary linear codes with prescribed minimum distance let n 8 d be the smallest integer n for which a binary linear code of length n dimension 8 and minimum distance d exists we prove that n 8 18 42 n 8 26 58 n 8 28 61 n 8 30 65 n 8 34 74 n 8 36 77 n 8 38 81 n 8 42 89 and n 8 60 124 after these results all values of n 8 d are known

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

from ds bounds for cyclic codes to true minimum distance for abelian codes in this paper we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets ds bounds to abelian or multivariate codes through the notion of mathbb b apparent distance we also study conditions for an abelian code to verify that its mathbb b apparent distance reaches its true minimum distance then we construct some codes as an application

trellis complexity of short linear codes an extended table of shuurman s bounds on the state complexity of short binary linear codes is presented some new lower and upper bounds are obtained most of the newly found codes are based on the so called double zero tail termination dzt construction

the twisted squaring construction trellis complexity and generalized weights of bch and qr codes the structure of the twisted squaring construction a generalization of the squaring construction is studied with respect to trellis diagrams and complexity we show that binary affine invariant codes which include the extended primitive bch codes and the extended binary quadratic residue codes are equivalent to twisted squaring construction codes in particular a recursive symmetric reversible design of the bch codes is derived using these constructions the parameters of the minimal trellis diagram of the bch codes are determined including the componentwise state space profile and trellis complexity new designs and permutations that yield low trellis complexity for the quadratic residue codes are presented generalized hamming weights are derived from these constructions as an example the 48 24 12 quadratic residue code is analyzed a strictly componentwise optimal permutation is derived and the corresponding state space profile and complete generalized hamming weight hierarchy are obtained

graphical codes revisited the set of all even subgraphs of a connected graph g on p vertices with q edges forms a binary linear code c c sub e g with parameters q q p 1 g where g is the girth of g such codes were studied systematically by bredeson and hakimi 1967 and hakimi and bredeson 1968 who were concerned with the problems of augmenting c to a larger q k g code and of efficiently decoding such augmented graphical codes we give a new approach to these problems by requiring the augmented codes to be graphical on one hand we present two construction methods which turn out to contain the methods proposed by hakimi and bredeson as special cases as we show this not only gives a better understanding of their construction it also results in augmenting codes of larger dimension we look at the case of 1 error correcting graphical codes in some detail in particular we show how to obtain the extended hamming codes as purely graphical codes by our approach on the other hand we follow a suggestion of ntafos and hakimi 1981 and use techniques from combinatorial optimization to give decoding procedures for graphical codes which turn out to be considerably more efficient than the approach via majority logic decoding proposed by bredeson and hakimi we also consider the decoding problem for the even graphical code based on the complete graph k sub 2n in more detail we discuss an efficient hardware implementation of an encoding decoding scheme for these codes and show that things may be arranged in such a way that one can also correct all adjacent double errors finally we discuss nonlinear graphical codes

some decomposable codes the a x b x a b x construction codes with decomposable structure allow the use of multistage decoding procedures to achieve suboptimum bounded distance error performance with reduced decoding complexity this correspondence presents some new decomposable codes including a class of distance 8 codes that are constructed based on the a x b x a b x construction method some existing best codes are shown to be decomposable and hence can be decoded with multistage decoding