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We consider deletion correcting codes over a \(q\) -ary alphabet. It is well known that any code capable of correcting \(s\) deletions can also correct any combination of \(s\) total insertions and deletions. To obtain asymptotic upper bounds on code size, we apply a packing argument to channels that perform different mixtures of insertions and deletions. Even though the set of codes is identical for all of these channels, the bounds that we obtain vary. Prior to this paper, only the bounds corresponding to the all-insertion case and the all-deletion case were known. We recover these as special cases. The bound from the all-deletion case, due to Levenshtein, has been the best known for more than 45 years. Our generalized bound is better than Levenshtein's bound whenever the number of deletions to be corrected is larger than the alphabet size. [ABSTRACT FROM AUTHOR]

Binary codes over 2D arrays are very useful in data storage, where each array column represents a storage device or unit that may suffer failure. In this paper, we propose a new framework for probabilistic construction of codes on 2D arrays. Instead of a pure combinatorial erasure model used in traditional array codes, we propose a mixed combinatorial-probabilistic model of limiting the number of column failures, and assuming a binary erasure channel in each failing column. For this model, we give code constructions and detailed analysis that allow sustaining a large number of column failures with graceful degradation in the fraction of erasures correctable in failing columns. Another advantage of the new framework is that it uses low-complexity iterative decoding. The key component in the analysis of the new codes is to analyze the decoding graphs induced by the failed columns, and infer the decoding performance as a function of the code design parameters, as well as the array size and failure parameters. A particularly interesting class of codes, called probabilistically maximum distance separable (MDS) array codes, gives fault-tolerance that is equivalent to traditional MDS array codes. The results also include a proof that the 2D codes outperform standard 1D low-density parity-check codes. [ABSTRACT FROM PUBLISHER]

In this paper, we study \BBZ2\BBZ4-additive cyclic codes. These codes are identified as \BBZ4[x]-submodules of the ring Rr,s=\BBZ2[x]/\langle x^{r}-1\rangle\times\BBZ4\left[x\right]/\langle x^{s}-1\rangle. The algebraic structure of this family of codes is studied and a set of generator polynomials for this family as a \BBZ4[x]-submodule of the ring Rr,s is determined. We show that the duals of \BBZ2\BBZ4-additive cyclic codes are also cyclic. We also present an infinite family of Maximum Distance separable with respect to the singleton bound codes. Finally, we obtain a number of binary linear codes with optimal parameters from the \BBZ2\BBZ4-additive cyclic codes. [ABSTRACT FROM AUTHOR]

A \BBZ2\BBZ4Q8-code is the binary image, after a Gray map, of a subgroup of \BBZ2^{k1}\times\BBZ4^{k2}\times Q8^{k3}, where Q8 is the quaternion group on eight elements. Such \BBZ2\BBZ4Q8-codes are translation invariant propelinear codes as are the well known \BBZ4-linear or \BBZ2\BBZ4-linear codes. In this paper, we show that there exist “pure” \BBZ2\BBZ4Q8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the \BBZ2\BBZ4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of \BBZ2\BBZ4Q8-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard. [ABSTRACT FROM AUTHOR]

Recent work by Divsalar has shown that properly designed protograph-based low-density parity-check codes typically have minimum (Hamming) distance linearly increasing with block length. This fact rests on ensemble arguments over all possible expansions of the base protograph. However, when implementation complexity is considered, the expansions are frequently selected from a smaller class of structured expansions. For example, protograph expansion by cyclically shifting connections generates a quasi-cyclic (QC) code. Other recent work by Smarandache and Vontobel has provided upper bounds on the minimum distance of QC codes. In this paper, we generalize these bounds to punctured QC codes and then show how to tighten these for certain classes of codes. We then evaluate these upper bounds for the family of protograph codes known as AR4JA codes that have been recommended for use in deep space communications in a standard established by the Consultative Committee for Space Data Systems. At block lengths larger than 4400 bits, these upper bounds fall well below the ensemble lower bounds. [ABSTRACT FROM AUTHOR]

This paper studies coding schemes for the q-ary symmetric channel based on binary low-density parity-check (LDPC) codes that work for any alphabet size q=2^m, m\in \BBN , thus complementing some recently proposed packet-based schemes requiring large q. First, theoretical optimality of a simple layered scheme is shown; then, a practical coding scheme based on a simple modification of standard binary LDPC decoding is proposed. The decoder is derived from first principles and using a factor-graph representation of a front end that maps q-ary symbols to groups of m bits connected to a binary code. The front end can be processed with a complexity that is linear in m=\log 2 q. An extrinsic information transfer chart analysis is carried out and used for code optimization. Finally, it is shown how the same decoder structure can also be applied to a larger class of q-ary channels. [ABSTRACT FROM PUBLISHER]

The first-order moment of (d,k)-constrained codes is investigated in this paper. A generalized moment balancing template is proposed to encode a (d,k) sequence into a single insertion or deletion correcting codeword without losing the constraint property. By relocating 0's in moment balancing runs, which appear in a pairwise manner of a (d,k) sequence, the first-order moment of this sequence can be modified to satisfy the Varshamov–Tenengolts construction. With a reasonably large base in the modulo system introduced by the Varshamov–Tenengolts construction, this generalized moment balancing template can be applied to run-length limited sequences. The asymptotic bound of the redundancy introduced by the template for (d,k) sequences is of the same order as the universal template for random sequences and, therefore, the redundancy is small and suitable for long sequences of practical interest. [ABSTRACT FROM AUTHOR]

In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal [38,19,8] self-dual codes, two s-extremal [38,19,6] codes, and 1730 s-extremal [38,19,8] codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of s-extremal [38,19,6] codes permits to achieve the classification of all s-extremal codes with d=6. [ABSTRACT FROM AUTHOR]

Nine new two-dimensional Optical Orthogonal Codes (2-D OOCs) are presented here, all sharing the common feature of a code size that is much larger in relation to the number of time slots than those of constructions appearing previously in the literature. Each of these constructions is either optimal or asymptotically optimal with respect to either the original Johnson bound or else a nonbinary version of the Johnson bound introduced in this paper. The first five codes are constructed using polynomials over finite fields—the first construction is optimal while the remaining four are asymptotically optimal. The next two codes are constructed using rational functions in place of polynomials and these are asymptotically optimal. The last two codes, also asymptotically optimal, are constructed by composing two of the above codes with a constant weight binary code. Also presented is a three-dimensional Optical Orthogonal Code (3-D OOC) that exploits the polarization dimension. Finally, phase-encoded optical CDMA is considered and construction of two efficient codes are provided. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there does not exist a binary self-dual doubly even [72, 36, 16] code with an automorphism of order 9. To do so, we apply a method for constructing binary self-dual codes possessing an automorphism of order p^2 for an odd prime p. [ABSTRACT FROM PUBLISHER]

Let m and k be positive integers with m/gcd(m,k) being odd, for a\in \BBR and b\in \BBL, the exponential sum \sumx\in \BBLi^{Tr(ax+2bx^{2^{k}+1})} is studied systematically in this paper, where i=\sqrt -1, \BBR =\BBG \BBR (4,m) is a Galois ring, \BBL is the Teichmüller set of \BBR and Tr(\cdot) is the trace function from the Galois ring \BBR to \BBZ4. Through the discussions on the solutions of certain equations and the newly developed theory of \BBZ4-valued quadratic forms, the distribution of the exponential sum is completely determined. As its applications, we can determine the Lee weight and Hamming weight distributions of a class of codes \cal C^k over \BBZ4 and the correlation distribution of a quaternary sequence family \cal U^k, respectively. Furthermore, the Hamming weight distributions of the binary codes obtained from \cal C^k under the most significant bit (MSB) and Gray maps are also determined. For the MSB map sequences of \cal U^k, the nontrivial maximal correlation value is given and the correlation distribution is determined for the Gray map sequences of \cal U^k. It should be noted that the distribution of the exponential sum for the case \gcd (m,k)\ne 1 is obtained for the first time, and then the corresponding codes and sequences are novel. [ABSTRACT FROM AUTHOR]