Searching on combinatorial structures and integer codes

SIAM Journal on Discrete Mathematics, 2010

One peculiarity with deletion-correcting codes is that perfect t-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius t with respect to the Levenshteȋn distance may be of different sizes. There is interest, therefore, in determining all possible sizes of a perfect t-deletion-correcting code, given the length n and the alphabet size q. In this paper, we determine completely the spectrum of possible sizes for perfect q-ary 1-deletion-correcting codes of length three for all q, and perfect q-ary 2-deletion-correcting codes of length four for almost all q, leaving only a small finite number of cases in doubt.

Designs, Codes and Cryptography, 2008

We investigate binary sequences which can be obtained by concatenating the columns of (0,1)-matrices derived from permutation sequences. We then prove that these binary sequences are subsets of a surprisingly diverse ensemble of codes, namely the Levenshtein codes, capable of correcting insertion/deletion errors; spectral null codes, with spectral nulls at certain frequencies; as well as being subsets of run-length limited codes, Nyquist null codes and constant weight codes.

IEEE Transactions on Information Theory, 1994

Properties of nonlinear perfect binary codes are investigated and several new constructions of perfect codes are derived from these properties. An upper bound on the cardinality of the intersection of two perfect codes of length n is presented, and perfect codes whose intersection attains the upper bound are constructed for all n. As an immediate consequence of the proof of the upper bound we obtain a simple closed-form expression for the weight distribution of a perfect code. Furthermore, we prove that the characters of a perfect code satisfy certain constraints, and provide a sufficient condition for a binary code to be perfect. The latter result is employed to derive a generalization of the construction of Phelps, which is shown to give rise to some perfect codes that are nonequivalent to the perfect codes obtained from the known constructions. Moreover, for any m 2 4 we construct jidl-rank perfect binary codes of length 2" -1. These codes are obviously nonequivalent to any of the previously known perfect codes. Furthermore the latter construction exhibits the existence of full-rank perfect tilings. Finally, we construct a set of 22c" nonequivalent perfect codes of length n, for sufficiently large n and a constant c = 0.5 -E. Precise enumeration of the number of codes in this set provides a slight improvement over the results previously reported by Phelps.

2006

Siam Journal on Discrete Mathematics, 2010

A set F of ordered k-tuples of distinct elements of an n-set is pairwise reverse free if it does not contain two ordered k-tuples with the same pair of elements in the same pair of coordinates in reverse order. Let F (n, k) be the maximum size of a pairwise reverse-free set. In this paper we focus on the case of 3-tuples and prove lim F (n, 3)/ n 3 = 5/4, more exactly, 5 24 n 3 − 1 2 n 2 − O(n log n) < F (n, 3) ≤ 5 24 n 3 − 1 2 n 2 + 5 8 n, and here equality holds when n is a power of 3. Many problems remain open.

Designs, Codes and …, 2001

The concept of diameter perfect codes, which seems to be a natural generalization of perfect codes (codes attaining the sphere-packing bound) is introduced. This was motivated by the "code-anticode" bound of Delsarte in distance regular graphs. This bound in conjunction with the recent complete solutions of diametric problems in the Hamming graph H q (n) and the Johnson graph J (n, k) gives a sharpening of the sphere-packing bound. Some necessary conditions for the existence of diameter perfect codes are given. In the Hamming graph all diameter perfect codes over alphabets of prime power size are characterized. The problem of tiling of the vertex set of J (n, k) with caps (and maximal anticodes) is also examined.

Discrete Mathematics, 2009

We replace the usual setting for error-correcting codes (i.e. vector spaces over finite fields) with that of permutation groups. We give an algorithm which uses a combinatorial structure which we call an uncovering-by-bases, related to covering designs, and construct some examples of these. We also analyse the complexity of the algorithm. We then formulate a conjecture about uncoverings-by-bases, for which we give some supporting evidence and prove for some special cases. In particular, we consider the case of the symmetric group in its action on 2-subsets, where we make use of the theory of graph decompositions. Finally, we discuss the implications this conjecture has for the complexity of the decoding algorithm.

Quantum Electronics, 2005

The algebraic solution of a`complex' problem of synthesis of phase-coded (PC) sequences with the zero level of side lobes of the cyclic autocorrelation function (ACF) is proposed. It is shown that the solution of the synthesis problem is connected with the existence of difference sets for a given code dimension. The problem of estimating the number of possible code combinations for a given code dimension is solved. It is pointed out that the problem of synthesis of PC sequences is related to the fundamental problems of discrete mathematics and, érst of all, to a number of combinatorial problems, which can be solved, as the number factorisation problem, by algebraic methods by using the theory of Galois éelds and groups.

2022 IEEE International Symposium on Information Theory (ISIT)

Motivated by applications to DNA-storage, flash memory, and magnetic recording, we study perfect burstcorrecting codes for the limited-magnitude error channel. These codes are lattices that tile the integer grid with the appropriate error ball. We construct two classes of such perfect codes correcting a single burst of length 2 for (1, 0)-limited-magnitude errors, both for cyclic and non-cyclic bursts. We also present a generic construction that requires a primitive element in a finite field with specific properties. We then show that in various parameter regimes such primitive elements exist, and hence, infinitely many perfect burst-correcting codes exist. Index Terms Integer coding, perfect codes, burst-correcting codes, lattices, limited-magnitude errors I. INTRODUCTION In many communication or storage systems, errors tend to occur in close proximity to each other, rather than occurring independently of each other. If the errors are confined to an interval of positions of length b, they are referred to as a burst of length b. Note that not all the positions in the interval are necessarily erroneous. A code that can correct any single burst of length b is called a b-burst-correcting code. The design of burst-correcting codes has been researched in the error models of substitutions, deletions and insertions. Concerning the substitutions, Abdel-Ghaffar et al. [1], [2] showed the existence of optimum cyclic bburst-correcting codes for any fixed b, and Etzion [10] gave a construction for perfect binary 2-burst-correcting codes. As for deletions and insertions, it has been shown in [20] that correcting a single burst of deletions is equivalent to correcting a single burst of insertions. Codes correcting a burst of exactly b consecutive deletions, or a burst of up to b consecutive deletions, were presented in [17], [20], with the redundancy being of optimal asymptotic order. The b-burst-correcting codes pertaining to deletions were treated in [3], called codes correcting localized deletions therein, and a class of such codes of asymptotically optimal redundancy was proposed. Similarly, permutation codes correcting a single burst of b consecutive deletions were studied in [8].

Theoretical Computer Science, 2005

The centralizer of a set of words X is the largest set of words C(X) commuting with X: XC(X) = C(X)X. It has been a long standing open question due to , whether the centralizer of any rational set is rational. While the answer turned out to be negative in general, see Kunc 2004, we prove here that the situation is different for codes: the centralizer of any rational code is rational and if the code is finite, then the centralizer is finitely generated. This result has been previously proved only for binary and ternary sets of words in a series of papers by the authors and for prefix codes in an ingenious paper by Ratoandromanana 1989 -many of the techniques we use in this paper follow her ideas. We also give in this paper an elementary proof for the prefix case.