3,473 research outputs found
Self-dual cyclic codes over finite chain rings
Let be a finite commutative chain ring with unique maximal ideal , and let be a positive integer coprime with the
characteristic of . In this paper, the algebraic
structure of cyclic codes of length over 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
The lengths of Hermitian Self-Dual Extended Duadic Codes
Duadic codes are a class of cyclic codes that generalizes quadratic residue
codes from prime to composite lengths. For every prime power q, we characterize
the integers n such that over the finite field with q^2 elements there is a
duadic code of length n having an Hermitian self-dual parity-check extension.
We derive using analytic number theory asymptotic estimates for the number of
such n as well as for the number of lengths for which duadic codes exist.Comment: To appear in the Journal of Pure and Applied Algebra. 21 pages and 1
Table. Corollary 4.9 and Theorem 5.8 have been added. Some small changes have
been mad
Application of Constacyclic codes to Quantum MDS Codes
Quantum maximal-distance-separable (MDS) codes form an important class of
quantum codes. To get -ary quantum MDS codes, it suffices to find linear MDS
codes over satisfying by the
Hermitian construction and the quantum Singleton bound. If
, we say that is a dual-containing code. Many new
quantum MDS codes with relatively large minimum distance have been produced by
constructing dual-containing constacyclic MDS codes (see \cite{Guardia11},
\cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study on
the existence condition for nontrivial dual-containing constacyclic codes. This
would help us to avoid unnecessary attempts and provide effective ideas in
order to construct dual-containing codes. Several classes of dual-containing
MDS constacyclic codes are constructed and their parameters are computed.
Consequently, new quantum MDS codes are derived from these parameters. The
quantum MDS codes exhibited here have parameters better than the ones available
in the literature.Comment: 16 page
Conference matrices and unimodular lattices
Conference matrices are used to define complex structures on real vector
spaces. Certain lattices in these spaces become modules for rings of quadratic
integers. Multiplication of these lattices by non-principal ideals yields
simple constructions of further lattices including the Leech lattice.Comment: 17 pages. Subitted to European Journal of Combinatoric
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