on the stopping redundancy of reed muller codes the stopping redundancy of the code is an important parameter which arises from analyzing the performance of a linear code under iterative decoding on a binary erasure channel in this paper we will consider the stopping redundancy of reed muller codes and related codes let r lscr m be the reed muller code of length 2m and order lscr schwartz and vardy gave a recursive construction of parity check matrices for the reed muller codes and asked whether the number of rows in those parity check matrices is the stopping redundancy of the codes we prove that the stopping redundancy of r m 2 m which is also the extended hamming code of length 2m is 2m 1 and thus show that the recursive bound is tight in this case we prove that the stopping redundancy of the simplex code equals its redundancy several constructions of codes for which the stopping redundancy equals the redundancy are discussed we prove an upper bound on the stopping redundancy of r 1 m this bound is better than the known recursive bound and thus gives a negative answer to the question of schwartz and vardy

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

an sp 2 covariant version of generalized canonical quantization of dynamical systems with linearly dependent constraints an sp 2 covariant version of the method of generalized canonical quantization of dynamical systems with linearly dependent first class constraints is proposed the existence theorem for solutions of generating equations of a gauge algebra is proved and the natural arbitrariness in these solutions is described the scheme proposed is shown to be equivalent to the standard version of generalized canonical quantization

improved upper bounds on stopping redundancy for a linear block code with minimum distance d its stopping redundancy is the minimum number of check nodes in a tanner graph representation of the code such that all nonempty stopping sets have size d or larger we derive new upper bounds on stopping redundancy for all linear codes in general and for maximum distance separable mds codes specifically and show how they improve upon previous results for mds codes the new bounds are found by upper bounding the stopping redundancy by a combinatorial quantity closely related to turan numbers the turan number t v k t is the smallest number of t subsets of a v set such that every k subset of the v set contains at least one of the t subsets asymptotically we show that the stopping redundancy of mds codes with length n and minimum distance d 1 is t n d 1 d 2 1 o n 1 for fixed d and is at most t n d 1 d 2 3 o n 1 for fixed code dimension k n d 1 for d 3 4 we prove that the stopping redundancy of mds codes is equal to t n d 1 d 2 for which exact formulas are known for d 5 we show that the stopping redundancy of mds codes is either t n 4 3 or t n 4 3 1

stability of a mixed functional equation in several variables on banach modules abstract in this paper we establish the general solution of the functional equation i 1 n f x i 1 n j 1 n x j i 1 n f x i n f 1 n i 1 n x i n 2 and use a fixed point method to prove its generalized hyers ulam stability in banach modules over a unital banach algebra

asymptotic enumeration methods for analyzing ldpc codes we show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of low density parity check ldpc codes in particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular ldpc code ensembles we then consider the binary erasure channel bec using these estimates we derive lower bounds on the error exponent under iterative decoding of ldpc codes used over the bec both regular and irregular code structures are considered these bounds are compared to the corresponding bounds when optimal maximum likelihood ml decoding is applied

on the stopping distance of array code parity check matrices for q an odd prime and 1 les m les q we study two binary qm times q2 parity check matrices for binary array codes for both parity check matrices we determine the stopping distance and the minimum distance of the associated code for 2 les m les 3 and for m q 4 5 in the case m q 4 7 the stopping distance and the related minimum distance are also determined for one of the given parity check matrices moreover we give a lower bound on the stopping distances for m 3 and q 3

parity check density versus performance of binary linear block codes over memoryless symmetric channels we derive lower bounds on the density of parity check matrices of binary linear codes which are used over memoryless binary input output symmetric mbios channels the bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity they are valid for every sequence of binary linear block codes if there exists a decoding algorithm under which the average bit error probability vanishes for every mbios channel we construct a sequence of ensembles of regular low density parity check ldpc codes so that an upper bound on the asymptotic density of their parity check matrices scales similarly to the lower bound the tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right regular ldpc codes which was introduced by shokrollahi and which is known to achieve the capacity of this channel under iterative message passing decoding we show that this sequence of ensembles is asymptotically optimal in a sense to be defined in this paper strengthening a result of shokrollahi finally we derive lower bounds on the bit error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs and study their performance limitations over mbios channels the latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error correction codes

on decoding of low density parity check codes over the binary erasure channel this paper investigates decoding of low density parity check ldpc codes over the binary erasure channel bec we study the iterative and maximum likelihood ml decoding of ldpc codes on this channel we derive bounds on the ml decoding of ldpc codes on the bec we then present an improved decoding algorithm the proposed algorithm has almost the same complexity as the standard iterative decoding however it has better performance simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm we also provide some graph theoretic properties of different decoding algorithms of ldpc codes over the bec which we think are useful to better understand the ldpc decoding methods in particular for finite length codes

computing the stopping distance of a tanner graph is np hard two decision problems related to the computation f stopping sets in tanner graphs are shown to be np complete it follows as a consequence that there exists no polynomial time algorithm for computing the stopping distance of a tanner graph unless p np

results on parity check matrices with optimal stopping and or dead end set enumerators the performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying tanner graph or equivalently the parity check matrix representing the code in this correspondence we introduce the notion of dead end sets to explicitly demonstrate this dependency the choice of the parity check matrix entails a tradeoff between performance and complexity we give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity check matrices further we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator

the decoding error probability of linear codes over the erasure channel in this paper we study the decoding error probability of linear codes over the erasure channel under the list decoding the notion of the q ell incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding by calculating the q ell incorrigible set distributions the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions for the ensemble of all n k linear codes the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined furthermore the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all n nr linear codes

parity check matrices separating erasures from errors most decoding algorithms of linear codes in general are designed to correct or detect errors however many channels cause erasures in addition to errors in principle decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code for any given linear code and any given maximum number of correctable erasures parity check matrices are introduced that yield parity check equations which do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures these matrices allow for the separation of erasures from errors to facilitate decoding several constructions of such separating parity check matrices are presented to reduce decoding complexity separating parity check matrices with small number of rows are preferred the minimum number of rows in a parity check matrix separating a given maximum number of erasures is called the separating redundancy upper and lower bounds on the separating redundancies are derived in particular it is shown that the separating redundancies tend to grow linearly with the number of rows in full rank parity check matrices of codes the separating redundancies of some classes of codes are determined for some maximum numbers of erasures

single exclusion number and the stopping redundancy of mds codes for a linear block code c its stopping redundancy is defined as the smallest number of check nodes in a tanner graph for c such that there exist no stopping sets of size smaller than the minimum distance of c schwartz and vardy conjectured that the stopping redundancy of a maximum distance separable mds code should only depend on its length and minimum distance we define the n t single exclusion number s n t as the smallest number of i subsets of an n set such that for each i subset of the n set i 1 t 1 there exists a i subset that contains all but one element of the i subset new upper bounds on the single exclusion number are obtained via probabilistic methods recurrent inequalities as well as explicit constructions the new bounds are used to better understand the stopping redundancy of mds codes in particular it is shown that for n k n d 1 d mds codes as n rarr infin the stopping redundancy is asymptotic to s n d 2 if d o radic n or if k o radic n k rarr infin thus giving partial confirmation of the schwartz vardy conjecture in the asymptotic sense

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

on the stopping distance and the stopping redundancy of codes it is now well known that the performance of a linear code copf under iterative decoding on a binary erasure channel and other channels is determined by the size of the smallest stopping set in the tanner graph for copf several recent papers refer to this parameter as the stopping distance s of copf this is somewhat of a misnomer since the size of the smallest stopping set in the tanner graph for copf depends on the corresponding choice of a parity check matrix it is easy to see that s les d where d is the minimum hamming distance of copf and we show that it is always possible to choose a parity check matrix for copf with sufficiently many dependent rows such that s d we thus introduce a new parameter the stopping redundancy of copf defined as the minimum number of rows in a parity check matrix h for copf such that the corresponding stopping distance s h attains its largest possible value namely s h d we then derive general bounds on the stopping redundancy of linear codes we also examine several simple ways of constructing codes from other codes and study the effect of these constructions on the stopping redundancy specifically for the family of binary reed muller codes of all orders we prove that their stopping redundancy is at most a constant times their conventional redundancy we show that the stopping redundancies of the binary and ternary extended golay codes are at most 34 and 22 respectively finally we provide upper and lower bounds on the stopping redundancy of mds codes

complete enumeration of stopping sets of full rank parity check matrices of hamming codes stopping sets and in particular their numbers and sizes play an important role in determining the performance of iterative decoders of linear codes over binary erasure channels in the 2004 shannon lecture mceliece presented an expression for the number of stopping sets of size three for a full rank parity check matrix of the hamming code in this correspondence we derive an expression for the number of stopping sets of any given size for the same parity check matrix

stopping set distributions of some reed muller codes stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel bec let c be a binary n k linear code with parity check matrix h where the rows of h may be dependent a stopping set s of c with parity check matrix h is a subset of column indices of h such that the restriction of h to s does not contain a row of weight one the stopping set distribution ti h i 0n enumerates the number of stopping sets with size i of c with parity check matrix h note that stopping sets and stopping set distribution are related to the parity check matrix h of c let h be the parity check matrix of c which is formed by all the nonzero codewords of its dual code c a parity check matrix h is called bec optimal if ti h ti h i 0 1 n and h has the smallest number of rows in this paper we study stopping sets stopping set distributions and bec optimal parity check matrices of binary linear codes using finite geometry in combinatorics we obtain bec optimal parity check matrices and then determine the stopping set distributions for the simplex codes the hamming codes the first order reed muller codes and the extended hamming codes which are some reed muller codes or their shortening or puncturing versions

more on the stopping and minimum distances of array codes for q an odd prime and 1 m q two specific binary qm q2 parity check matrices denoted by hp m q and hi m q are considered the corresponding binary codes cp m q and ci m q respectively are called proper and improper array codes with parameters m and q given a parity check matrix h representing a binary code c let s h denote the stopping distance of h and d c be the minimum hamming distance of c it is known that that s hi m q s hp m q d ci m q d cp m q for m 3 in this paper we show that these equalities do not hold for all values of m and q in particular although s hp 4 7 d cp 4 7 8 we have s hi 4 7 9 and d ci 4 7 10 it is also shown that s hp 5 1 dcp 5 11 10 while s hi 5 11 11 and d ci 5 11 12 this suggests that in many cases the improper array codes would perform better than the proper array codes over the awgn and binary erasure channels performance results are given which confirm this claim the combinatorial structure of the eight element stopping sets for h m 4 q 5 is also determined

stopping set elimination by parity check matrix extension via integer linear programming error rate floor phenomenon is known to be a serious impediment to the use of low density parity check ldpc codes for some practical applications that demand high data reliability in the case of binary erasure channels becs certain error prone patterns known as stopping sets are proven to cause this performance degradation a possible approach to diminish this drawback over becs is to eliminate stopping sets by parity check matrix extension given a parity check matrix h and a list mathcal l of its stopping sets we present an integer linear programming ilp formulation to find a parity check equation which eliminates the maximum number of stopping sets in mathcal l one of the distinguishing advantages of the proposed scheme is its flexibility for modifications such as limiting the weight of the new parity check row making the new row redundant or linearly independent 4 cycle avoidance and taking into account the sizes of stopping sets armed with these adjustments the method can provide good performance improvements as evidenced by simulation results furthermore for a given varrho in mathbb n by extending the basic formulation we provide an ilp formulation for finding a set of size varrho of parity check equations which can best eliminate the stopping sets in mathcal l among all such sets

on parity check collections for iterative erasure decoding that correct all correctable erasure patterns of a given size recently there has been interest in the construction of small parity check sets for iterative decoding of the hamming code with the property that each uncorrectable or stopping set of size three is the support of a codeword and hence uncorrectable anyway here we reformulate and generalize the problem and improve on this construction we show that a parity check collection that corrects all correctable erasure patterns of size m for the hamming code with codimension r provides in fact for all codes of codimension r a corresponding generic parity check collection with this property this leads in a natural way to a necessary and sufficient condition for such generic parity check collections we use this condition to construct a generic parity check collection for codes of codimension r correcting all correctable erasure patterns of size at most m for all r and mlesr thus generalizing the known construction for m 3 then we discuss optimality of our construction and show that it can be improved for mges3 and r large enough finally we discuss some directions for further research