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The theory of cyclotomy dates back to Gauss and has a number of applications in combinatorics, coding theory, and cryptography. Cyclotomy over a residue class ring \BBZv can be divided into classical cyclotomy or generalized cyclotomy, depending on v prime or composite. In this paper, we introduce a generalized cyclotomy of order d over \BBZp1^{e1p2^{e2},\ldots, pn^{en}}, which includes Whiteman's and Ding-Helleseth's generalized cyclotomy as special cases. Here, p1,p2,\ldots,pn are pairwise distinct odd primes satisfying d\vert (pi-1) for all 1\leq i\leq n and e1,e2,\ldots,en are positive integers. We derive some basic properties of the corresponding cyclotomic numbers and obtain a general formula to compute them via classical cyclotomic numbers. As applications, we completely solve an open problem and a conjecture on Whiteman's generalized cyclotomy of order four over \BBZp1p2. Besides, we also construct an infinite series of near-optimal codebooks over \BBZp1p2, as well as some infinite series of asymptotically optimal difference systems of sets over \BBZp1^{e1p2^{e2},\ldots,pn^{en}}. [ABSTRACT FROM AUTHOR]

Various kinds of secure codes and their related hash families are broadly studied combinatorial structures for protecting copyrighted materials. The codewords in such a structure can be regarded as a subset of Q^N , the set of all $q$ -ary vectors of given length $N$ , satisfying some constraints. We use a hypergraph model to characterize the combinatorial structure. By applying a result of Duke et al. on the lower bound of the independence number of hypergraphs, we provide a new approach to evaluate the lower bounds for several kinds of secure codes and related hash families. In particular, the general method is illustrated via the examples of existence results on some perfect hash families, frameproof codes, and separable codes. [ABSTRACT FROM PUBLISHER]

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of Yang et al. into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeros. [ABSTRACT FROM PUBLISHER]

Permutation codes have been shown to be useful in power line communications, block ciphers, and multilevel flash memory models. Construction of such codes is extremely difficult. In fact, the only general lower bound known is the Gilbert–Varshamov type bound. In this paper, we establish a connection between permutation codes and independent sets in certain graphs. Using the connection, we improve the Gilbert–Varshamov bound asymptotically by a factor \log (n), when the code length n goes to infinity. [ABSTRACT FROM PUBLISHER]

Cyclic codes are widely employed in communication systems, storage devices, and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over \mathrm GF(q) with length n=(q^m-1)/(q-1) . Little is known about this class of BCH codes when $q>2$ . The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length n=(3^{m}-1)/2 are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated. [ABSTRACT FROM PUBLISHER]

Multiply constant-weight codes (MCWCs) were introduced recently to improve the reliability of certain physically unclonable function response. In this paper, the bounds of MCWCs and the constructions of optimal MCWCs are studied. First, we derive three different types of upper bounds which improve the Johnson-type bounds given by Chee et al. for some parameters. The asymptotic lower bound of MCWCs is also examined. Then, we obtain the asymptotic existence of two classes of optimal MCWCs, which shows that the Johnson-type bounds for MCWCs with distances 2\sum i=1^{m}wi-2 or $2mw-2w$ are asymptotically exact. Finally, we construct a class of optimal MCWCs with total weight four and distance six by establishing the connection between such MCWCs and a new kind of combinatorial structures. As a consequence, the maximum sizes of MCWCs with total weight less than or equal to four are determined almost completely. [ABSTRACT FROM AUTHOR]

The determination of weight distribution of cyclic codes involves the evaluation of Gauss sums and exponential sums. Despite some cases where a neat expression is available, the computation is generally rather complicated. In this note, we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros. This goal is achieved by building an unexpected connection between the corresponding exponential sums and the spectra of Hermitian forms graphs. [ABSTRACT FROM PUBLISHER]