Generalizing Bounds on the Minimum Distance of Cyclic Codes Using Cyclic Product Codes

Two generalizations of the Hartmann-Tzeng (HT) bound on the minimum distance of q-ary cyclic codes are proposed. The first one is proven by embedding the given cyclic code into a cyclic product code. Furthermore, we show that unique decoding up to this bound is always possible and outline a quadratic-time syndrome-based error decoding algorithm. The second bound is stronger and the proof is more involved. Our technique of embedding the code into a cyclic product code can be applied to other bounds, too and therefore generalizes them

Efficient Decoding of (binary) Cyclic Codes beyond the correction capacity of the code using Gröbner bases

The problem of decoding cyclic codes can be rewritten into an algebraic system of equations, whose solutions are closely related to the error that occured. Extensive work has been done previously, where it has been shown that the computation of a Gröbner basis of this algebraic system enables to decode up to the true minimum distance. The Gröbner basis computation can be done either as a preprocessing (formal decoding), with the parameters taken as variables, or for each word to be decoded (online decoding), with the parameters computed from the word and substituted into the system. For the formal decoding, it has been shown that decoding formulas for the coefficients of the locator polynomial are obtained from the formal Gröbner basis. Unfortunately, it becomes quickly impossible to compute this formal Gröbner basis even for codes of small length. Motivated by the problem of decoding quadratic residue (QR) codes, for which no general decoding algorithm is known, we improve on several points. First we introduce modified systems, without high degree equations, for which the Gröbner basis computation is easier. This enables to compute the formal Gröbner basis for longer codes. We show on the example of the [41,21,9] QR code that the formulas become quickly of large size, thus being useless for decoding. This indicates that the effort on the algebraic decoding of cyclic codes by formulas hits a wall. The other approach (online decoding) is more efficient from a computational point of view. Using general compilation methods for systems with parameters, we improve the efficiency of the computation. Many examples are given (for BCH codes of length 75, 511, for QR codes of length 41, 73, 89, 113 and for a code of length 75 which does not belong to a known class of codes). This method for decoding cyclic codes with Gröbner basis works for any cyclic codes, is automatic and enables to decode beyond the true minimum distance

Cyclic Orbit Codes

In network coding a constant dimension code consists of a set of k-dimensional subspaces of F_q^n. Orbit codes are constant dimension codes which are defined as orbits of a subgroup of the general linear group, acting on the set of all subspaces of F_q^n. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper we give a classification of cyclic orbit codes and propose a decoding procedure for a particular subclass of cyclic orbit codes.Comment: submitted to IEEE Transactions on Information Theor

Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes

We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on Information Theor

Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance

The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding radius based on an associated folded cyclic code. Furthermore, we present a third technique for bounding the minimum Hamming distance based on the embedding of a given repeated-root cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic burst error decoding procedure based on the third bound is outlined. Index Terms Bound on the minimum distance, burst error, efficient decoding, folded code, repeated-root cyclic code, repeated-root cyclic product cod

Optimal locally repairable codes of distance 33 and 44 via cyclic codes

Like classical block codes, a locally repairable code also obeys the Singleton-type bound (we call a locally repairable code {\it optimal} if it achieves the Singleton-type bound). In the breakthrough work of \cite{TB14}, several classes of optimal locally repairable codes were constructed via subcodes of Reed-Solomon codes. Thus, the lengths of the codes given in \cite{TB14} are upper bounded by the code alphabet size $q$. Recently, it was proved through extension of construction in \cite{TB14} that length of $q$-ary optimal locally repairable codes can be $q+1$ in \cite{JMX17}. Surprisingly, \cite{BHHMV16} presented a few examples of $q$-ary optimal locally repairable codes of small distance and locality with code length achieving roughly $q^2$. Very recently, it was further shown in \cite{LMX17} that there exist $q$-ary optimal locally repairable codes with length bigger than $q+1$ and distance propositional to $n$. Thus, it becomes an interesting and challenging problem to construct new families of $q$-ary optimal locally repairable codes of length bigger than $q+1$. In this paper, we construct a class of optimal locally repairable codes of distance $3$ and $4$ with unbounded length (i.e., length of the codes is independent of the code alphabet size). Our technique is through cyclic codes with particular generator and parity-check polynomials that are carefully chosen