Constacyclic codes over finite commutative chain rings

Abstract

Constructing codes that are easy to encode and decode, can detect and correct many errors and have a sufficiently large number of codewords is the primary aim of coding theory. Several metrics (e.g. Hamming metric, Lee metric, RT metric, etc.) have been introduced to study error-detecting and error-correcting properties of a code with respect to various communication channels. Among the prevalent metrics in coding theory, the Hamming metric is the most studied metric and it is suitable for orthogonal modulated channels. Singleton [74] derived the upper bound (called the Singleton bound) on the size of an arbitrary block code with respect to the Hamming metric. Linear codes that attain the Singleton bound are called maximum distance separable (MDS) Hamming codes. Later, motivated by the problem to transmit messages over several parallel communication channels with some channels not available for transmission, a non-Hamming metric, called the Rosenbloom-Tsfasman metric (or RT metric), was introduced by Rosenbloom and Tsfasman [70]; they also derived Singleton bound for codes with respect to the RT metric. Linear codes that attain the Singleton bound for the RT metric are called maximum distance separable (MDS) RT codes. Recently, Cassuto and Blaum [12, 13] established a new coding framework for channels whose outputs are overlapping pairs of symbols. Such channels are called symbol-pair read channels and the corresponding metric is called the symbol-pair metric. These channels are more suitable for high density data storage systems in which the spatial resolution of the reader is insufficient to isolate adjacent symbols. Chee et al. [15] derived a Singleton-type bound for codes with respect to the symbol-pair metric and constructed many maximum distance separable (MDS) symbol-pair codes, i.e., the codes attaining the Singleton-type bound with respect to the symbol-pair metric. Recently, Yaakobi et al. [82] extended the framework of symbol-pair read channels to b-symbol read channels, whose outputs are consecutive sequences of b _ 3 symbols. The corresponding metric is called the b-symbol metric. In a recent work, Ding et al. [28] derived a Singleton-type bound for codes over finite fields with respect to the b-symbol metric. The codes that attain the Singleton-type bound with respect to the b-symbol metric are called MDS b-symbol codes. MDS codes have the highest possible error-detecting and error-correcting capabilities for given code length, code size and alphabet size, hence they are considered optimal codes in that sense. Thus it is of great interest to study and find MDS codes with respect to various metrics. The derivative is a well-known operator of sequences and is useful in investigating the linear complexity of sequences in game theory, communication theory and cryptography (see [6, 14, 33, 83]). Etzion [32] first applied the derivative operator on codewords of linear codes over finite fields, and defined the depth of a codeword in terms of the derivative operator. He showed that there are exactly k distinct non-zero depths attained by non-zero codewords of a k-dimensional linear code, and that any k non-zero codewords with distinct depths form a basis of the code. This shows that the depth distribution is an interesting parameter of linear codes. In this thesis, we study algebraic structures of repeated-root constacyclic codes over finite commutative chain rings and their dual codes. We also explicitly determine Hamming distances, symbol-pair distances, b-symbol distances, RT distances, and RT weight distributions of several classes of repeatedroot constacyclic codes over finite commutative chain rings. Using these results, we identify several isodual, MDS Hamming, MDS RT, MDS symbol-pair and MDS b-symbol codes within the family of repeated-root constacyclic codes over finite commutative chain rings. We also discuss a decoding algorithm for repeated-root constacyclic codes of prime power lengths over finite commutative chain rings with respect to Hamming, symbol-pair and RT metrics. We also study depths of codewords of a class of repeated-root constacyclic codes over finite commutative chain rings. As a consequence, we explicitly determine depth distributions of this particular class of constacyclic codes over finite commutative chain rings. We also introduce two new turn-based roulette games and discuss their winning strategies by applying our results on depths of codewords of repeated-root constacyclic codes over finite commutative chain rings. These results are useful in encoding and decoding these codes and in studying their error-detecting and error-correcting capabilities with respect to various communication channels.