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

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

a directed search approach for unit memory convolutional codes a set of heuristic algorithms to numerically search for binary unit memory convolutional codes umc are presented along with a large number of new codes for 2 spl les k spl les 8 and code rate 1 4 spl les r 1 combinatorial optimization is used which involves selecting and then pairwise matching column vectors of the two n k umc tap weight matrices the column selection problem is that of finding the best 2n k binary linear block code bc in this correspondence the best bc generator matrix g is found by successively refining g using directed local exhaustive searches in particular the set of minimum weight codewords are used to find a subset of g to exhaustively search the umc search strategy pairwise matching problem uses a directed local exhaustive search similar to the bc directed search by using the concept of the terminated bc of the umc the heuristic algorithms developed in this correspondence are very robust and converge relatively quickly to the optimal or near optimal umc in addition although it is generally possible to achieve the block code upper bound for free distance we give a class of umcs which cannot achieve this bound

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

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

bounds on the free distance of systematic convolutional codes asymptotic bounds on the free distance of binary time varying systematic convolutional codes are developed from a unified point of view it is found that this unified approach provides some insight into the relationship between the distances of systematic versus nonsystematic codes it is also helpful from a constructive point of view the free distance bounds of systematic unit memory codes are also discussed

low complexity decoding of partial unit memory codes on precoded partial response channels cosets of convolutional codes can be used to obtain large free euclidean distances and short maximum zero run lengths at the output of the 1 d partial response channel prc we present a new soft decision decoding technique for cosets of convolutional codes on the preceded 1 d prc the decoding technique is especially well suited for cosets of partial unit memory pum convolutional codes a connection between the decoder trellises for cosets of block codes and pum codes on the 1 d prc is exploited to obtain a small number of operations per decoded information bit we prove that the new decoding technique needs fewer operations than the viterbi algorithm for any n n r r spl ges 1 pum code coset with n spl les 2 sup v r where v is the constraint length it is also indicated how to prove the same result for other classes of codes for many of the best known pum code cosets the new decoding technique requires both fewer operations per decoded information bit and smaller path memories than the viterbi algorithm needs for comparable cosets of punctured convolutional 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

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

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

state space realizations and monomial equivalence for convolutional codes we will study convolutional codes with the help of state space realizations it will be shown that two such minimal realizations belong to the same code if and only if they are equivalent under the full state feedback group this result will be used in order to prove that two codes with positive forney indices are monomially equivalent if and only if they share the same adjacency matrix the adjacency matrix is an invariant of the code obtained via a minimal state space realization and counts in a detailed way the weights of all possible outputs it contains full information about the weights of the codewords in the given code

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

generalized bootstrap for estimating equations we introduce a generalized bootstrap technique for estimators obtained by solving estimating equations some special cases of this generalized bootstrap are the classical bootstrap of efron the delete d jackknife and variations of the bayesian bootstrap the use of the proposed technique is discussed in some examples distributional consistency of the method is established and an asymptotic representation of the resampling variance estimator is obtained

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

concatenated codes with convolutional inner codes the minimum distance of concatenated codes with reed solomon outer codes and convolutional inner codes is studied for suitable combinations of parameters the minimum distance can be lower bounded by the product of the minimum distances of the inner and outer codes for a randomized ensemble of concatenated codes a lower bound of the gilbert varshamov type is proved

constante de bers en genre 2 we introduce a new tool the contiguity graph which enables us to determine the bers s constant in genus two

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

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

bounded distance decoding of unit memory codes we discuss minimum distance decoding of convolutional codes the relevant distance functions are defined and the set of correctable error patterns is described by a sequence of weight constraints decoding methods for error patterns of bounded weight are described and it is demonstrated that these methods offer a favorable combination of performance and complexity exact values and upper bounds on the error probability are calculated from finite state models of the decoding process

are discoveries spurious distributions of maximum spurious correlations and their applications over the last two decades many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool can the discoveries by such data mining approaches be spurious due to high dimensionality and limited sample size can our fundamental assumptions on exogeneity of covariates needed for such variable selection be validated with the data to answer these questions we need to derive the distributions of the maximum spurious correlations given certain number of predictors namely the distribution of the correlation of a response variable y with the best s linear combinations of p covariates x even when x and y are independent when the covariance matrix of x possesses the restricted eigenvalue property we derive such distributions for both finite s and diverging s using gaussian approximation and empirical process techniques however such a distribution depends on the unknown covariance matrix of x hence we use the multiplier bootstrap procedure to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach the results are further extended to the situation where residuals are from regularized fits our approach is then applied to construct the upper confidence limit for the maximum spurious correlation and testing exogeneity of covariates the former provides a baseline for guarding against false discoveries due to data mining and the latter tests whether our fundamental assumptions for high dimensional model selection are statistically valid our techniques and results are illustrated by both numerical examples and real data analysis

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

a class of one dimensional mds convolutional codes a class of one dimensional convolutional codes will be presented they are all mds codes i e have the largest distance among all one dimensional codes of the same length n and overall constraint length furthermore their extended row distances are computed and they increase with slope at least n in certain cases of the algebraic parameters we will also derive parity check matrices of vandermonde type for these codes finally cyclicity in the convolutional sense of 8 will be discussed for our class of codes it will turn out that they are cyclic if and only if the field element used in the generator matrix has order n this can be regarded as a generalization of the block code case

construction of partial unit memory mds convolutional codes maximum distance separable mds convolutional codes form an optimal family of convolutional codes the study of which is of great importance there are very few general algebraic constructions of mds convolutional codes in this paper we construct a large family of partial unit memory mds convolutional codes over mathbb f q with flexible parameters compared with the previous work the field size q required to define these codes is much smaller the construction also leads to many new strongly mds convolutional codes an important subclass of mds convolutional codes some examples are presented at the end of this paper

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

bounded minimum distance decoding of unit memory codes we propose an algorithm for bounded minimum distance decoding of partial unit memory codes up to half the designed extended row distance it makes use of a reduced trellis with the nodes found by bounded minimum distance decoding of the block codes used in the unit memory code the results can be extended to general multimemory codes the complexity of this algorithm is upper bounded by 2 d sub 1sup r 2d sub spl alpha times the complexity of the bounded minimum distance decoder of the block codes in the unit memory code here d sub spl alpha is the linear increase of the designed extended row distance d sub isup r

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

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

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

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

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

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

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

strongly mds convolutional codes maximum distance separable mds convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree in this paper a class of mds convolutional codes is introduced whose column distances reach the generalized singleton bound at the earliest possible instant such codes are called strongly mds convolutional codes they also have a maximum or near maximum distance profile the extended row distances of these codes will also be discussed briefly

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

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

one step sparse estimates in nonconcave penalized likelihood models fan and li propose a family of variable selection methods via penalized likelihood using concave penalty functions the nonconcave penalized likelihood estimators enjoy the oracle properties but maximizing the penalized likelihood function is computationally challenging because the objective function is nondifferentiable and nonconcave in this article we propose a new unified algorithm based on the local linear approximation lla for maximizing the penalized likelihood for a broad class of concave penalty functions convergence and other theoretical properties of the lla algorithm are established a distinguished feature of the lla algorithm is that at each lla step the lla estimator can naturally adopt a sparse representation thus we suggest using the one step lla estimator from the lla algorithm as the final estimates statistically we show that if the regularization parameter is appropriately chosen the one step lla estimates enjoy the oracle properties with good initial estimators computationally the one step lla estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood we conduct some monte carlo simulation to assess the finite sample performance of the one step sparse estimation methods the results are very encouraging

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

bounds on distances and error exponents of unit memory codes binary unit memory codes originally introduced by lee are investigated a few examples of constant unit memory codes are given and bounds on the distance profile and the free distances are discussed for time varying codes asymptotic lower bounds on the distance profile and the free distance are given the error probability for the codes used on a memoryless binary input output symmetric channel is asymptotically upper bounded the asymptotic results for the free distance and the error probability which are in some respects better than for conventional convolutional codes are interpreted by forney s inverse concatenation construction

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

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

comment on a tuning free robust and efficient approach to high dimensional regression consider the linear model 1 y x 0 e where y y 1 y n r n is a response vector x x 1 x n r n p is a design matrix 0 r p is an unknown coefficient vec

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

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 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

guarding against spurious discoveries in high dimensions many data mining and statistical machine learning algorithms have been developed to select a subset of covariates to associate with a response variable spurious discoveries can easily arise in high dimensional data analysis due to enormous possibilities of such selections how can we know statistically our discoveries better than those by chance in this paper we define a measure of goodness of spurious fit which shows how good a response variable can be fitted by an optimally selected subset of covariates under the null model and propose a simple and effective lamm algorithm to compute it it coincides with the maximum spurious correlation for linear models and can be regarded as a generalized maximum spurious correlation we derive the asymptotic distribution of such goodness of spurious fit for generalized linear models and l1 regression such an asymptotic distribution depends on the sample size ambient dimension the number of variables used in the fit and the covariance information it can be consistently estimated by multiplier bootstrapping and used as a benchmark to guard against spurious discoveries it can also be applied to model selection which considers only candidate models with goodness of fits better than those by spurious fits the theory and method are convincingly illustrated by simulated examples and an application to the binary outcomes from german neuroblastoma trials

on the macwilliams identity for convolutional codes the adjacency matrix associated with a convolutional code collects in a detailed manner information about the weight distribution of the code a macwilliams identity conjecture stating that the adjacency matrix of a code fully determines the adjacency matrix of the dual code will be formulated and an explicit formula for the transformation will be stated the formula involves the macwilliams matrix known from complete weight enumerators of block codes the conjecture will be proven for the class of convolutional codes where either the code itself or its dual does not have forney indices bigger than one for the general case the conjecture is backed up by many examples and a weaker version will be established

a macwilliams identity for convolutional codes the general case a macwilliams identity for convolutional codes will be established it makes use of the weight adjacency matrices of the code and its dual based on state space realizations the controller canonical form of the codes in question the macwilliams identity applies to various notions of duality appearing in the literature on convolutional coding theory

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

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

on the weight distribution of convolutional codes detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a minimal realization of the code we will show that this matrix is an invariant of the code moreover it will be proven that codes with the same adjacency matrix have the same dimension and the same forney indices and finally that for one dimensional binary convolutional codes the adjacency matrix determines the code uniquely up to monomial equivalence

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

active distances for convolutional codes a family of active distance measures for general convolutional codes is defined these distances are generalizations of the extended distances introduced by thommesen and justesen 1983 for unit memory convolutional codes it is shown that the error correcting capability of a convolutional code is determined by the active distances the ensemble of periodically time varying convolutional codes is defined and lower bounds on the active distances are derived for this ensemble the active distances are very useful in the analysis of concatenated convolutional encoders

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

unit memory codes for continuous phase modulation in an awgn channel the combination of unit memory codes and continuous phase modulation in an additive white gaussian noise channel is considered our motivation is based on the fact that when rouanne and costello s 1988 free distance lower bound is applied to unit memory codes an improved lower bound is obtained specific constructions are considered for 4 level modulation schemes with full response and weak partial response heuristic design rules are used for defining the subsets of codes within which an exhaustive search is conducted the coding gains of the best codes within these subsets are better than those of best previously known multi memory codes

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

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

convolutional codes in rank metric with application to random network coding random network coding recently attracts attention as a technique to disseminate information in a network this paper considers a noncoherent multishot network where the unknown and time variant network is used several times in order to create dependence between the different shots particular convolutional codes in rank metric are used these codes are so called partial unit memory p um codes i e convolutional codes with memory one first distance measures for convolutional codes in rank metric are shown and two constructions of p um codes in rank metric based on the generator matrices of maximum rank distance codes are presented second an efficient error erasure decoding algorithm for these codes is presented its guaranteed decoding radius is derived and its complexity is bounded finally it is shown how to apply these codes for error correction in random linear and affine network coding