Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes

2013-01-01
Lopez-Permouth, Sergio R.
Ozadam, Hakan
Özbudak, Ferruh
SZABO, Steve
Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p(a), m) and generating sets for its ideals are considered. It is shown that these generating sets are strong Groebner bases. A method for finding such sets in the case that a = 2 is given. This explicitly gives the Hamming distance of all cyclic codes of length p(s) over GR(p(2), m). The Hamming distance of certain constacyclic codes of length eta p(s) over F-pm is computed. A method, which determines the Hamming distance of the constacyclic codes of length eta p(s) over GR(p(a), m), where (eta, p) = 1, is described. In particular, the Hamming distance of all cyclic codes of length p(s) over GR(p(2), m) and all negacyclic codes of length 2p(s) over F-pm is determined explicitly. (c) 2012 Elsevier Inc. All rights reserved.
FINITE FIELDS AND THEIR APPLICATIONS

Suggestions

Spatially Coupled Codes Optimized for Magnetic Recording Applications
Esfahanizadeh, Homa; Hareedy, Ahmed; Dolecek, Lara (2017-02-01)
© 1965-2012 IEEE.Spatially coupled (SC) codes are a class of sparse graph-based codes known to have capacity-approaching performance. SC codes are constructed based on an underlying low-density parity-check (LDPC) code, by first partitioning the underlying block code and then putting replicas of the components together. Significant recent research efforts have been devoted to the asymptotic, ensemble-averaged study of SC codes, as these coupled variants of the existing LDPC codes offer excellent properties....
Construction of quasi-cyclic self-dual codes
Çomak, Pınar; Özbudak, Ferruh; Kim, Jon-Lark; Department of Cryptography (2013)
Quasi-cyclic and self-dual codes are interesting classes of linear codes. Quasi-cyclic codes are linear codes which takes maximum possible value of minimum distance among the codes with the same length and same dimension. Another class of interesting linear codes is the self-dual codes. Self-dual codes have close connections with group theory, lattice theory and design theory. There has been an active research on the classi fication of self-dual codes over fi nite fi elds and over rings. We study on constru...
The Concatenated Structure of Quasi-Cyclic Codes and an Improvement of Jensen's Bound
Guneri, Cem; Özbudak, Ferruh (2013-02-01)
Following Jensen's work from 1985, a quasi-cyclic code can be written as a direct sum of concatenated codes, where the inner codes are minimal cyclic codes and the outer codes are linear codes. We observe that the outer codes are nothing but the constituents of the quasi-cyclic code in the sense of Ling-Sole. This concatenated structure enables us to recover some earlier results on quasi-cyclic codes in a simple way, including one of our recent results which says that a quasi-cyclic code with cyclic constit...
On Linear Complementary Pairs of Codes
CARLET, Claude; Guneri, Cem; Özbudak, Ferruh; Ozkaya, Buket; SOLE, Patrick (Institute of Electrical and Electronics Engineers (IEEE), 2018-10-01)
We study linear complementary pairs (LCP) of codes (C, D), where both codes belong to the same algebraic code family. We especially investigate constacyclic and quasicyclic LCP of codes. We obtain characterizations for LCP of constacyclic codes and LCP of quasi-cyclic codes. Our result for the constacyclic complementary pairs extends the characterization of linear complementary dual (LCD) cyclic codes given by Yang and Massey. We observe that when C and I) are complementary and constacyclic, the codes C and...
New cubic self-dual codes of length 54, 60 and 66
Comak, PINAR; Kim, Jon Lark; Özbudak, Ferruh (2018-08-01)
We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length with the algebraic approach of Ling and Sol, (IEEE Trans Inf Theory 47(7):2751-2760, 2001. doi:. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66.
Citation Formats
S. R. Lopez-Permouth, H. Ozadam, F. Özbudak, and S. SZABO, “Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes,” FINITE FIELDS AND THEIR APPLICATIONS, pp. 16–38, 2013, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/38207.