1,040 research outputs found

    Construction of Quasi-Cyclic Product Codes

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    Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gr{\"o}bner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an explicit expression of the basis of the generating set of the quasi-cyclic product code is given. Furthermore, the reduced Gr{\"o}bner basis of a one-level quasi-cyclic product code is derived.Comment: 10th International ITG Conference on Systems, Communications and Coding (SCC), Feb 2015, Hamburg, German

    Self-Dual and Complementary Dual Abelian Codes over Galois Rings

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    Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring GR(pr,s)[G]{\rm GR}(p^r,s)[G], where GG is a finite abelian group and GR(pr,s){\rm GR}(p^r,s) is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in GR(pr,s)[G]{\rm GR}(p^r,s)[G]. A general formula for the number of such self-dual codes is established. In the case where gcd(G,p)=1\gcd(|G|,p)=1, the number of self-dual abelian codes in GR(pr,s)[G]{\rm GR}(p^r,s)[G] is completely and explicitly determined. Applying known results on cyclic codes of length pap^a over GR(p2,s){\rm GR}(p^2,s), an explicit formula for the number of self-dual abelian codes in GR(p2,s)[G]{\rm GR}(p^2,s)[G] are given, where the Sylow pp-subgroup of GG is cyclic. Subsequently, the characterization and enumeration of complementary dual abelian codes in GR(pr,s)[G]{\rm GR}(p^r,s)[G] are established. The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.Comment: 22 page

    Improved Lower Bounds for Constant GC-Content DNA Codes

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    The design of large libraries of oligonucleotides having constant GC-content and satisfying Hamming distance constraints between oligonucleotides and their Watson-Crick complements is important in reducing hybridization errors in DNA computing, DNA microarray technologies, and molecular bar coding. Various techniques have been studied for the construction of such oligonucleotide libraries, ranging from algorithmic constructions via stochastic local search to theoretical constructions via coding theory. We introduce a new stochastic local search method which yields improvements up to more than one third of the benchmark lower bounds of Gaborit and King (2005) for n-mer oligonucleotide libraries when n <= 14. We also found several optimal libraries by computing maximum cliques on certain graphs.Comment: 4 page

    Self-dual cyclic codes over finite chain rings

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    Let RR be a finite commutative chain ring with unique maximal ideal γ\langle \gamma\rangle, and let nn be a positive integer coprime with the characteristic of R/γR/\langle \gamma\rangle. In this paper, the algebraic structure of cyclic codes of length nn over RR is investigated. Some new necessary and sufficient conditions for the existence of nontrivial self-dual cyclic codes are provided. An enumeration formula for the self-dual cyclic codes is also studied.Comment: 15 page
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