On quasi-cyclic codes as a generalization of cyclic codes
Copyright policy )On quasi-cyclic codes as a generalization of cyclic codes
In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic ones such as generator polynomials and ideals. Indeed we show a one-to-one correspondence between l-quasi-cyclic codes of length m and ideals of M_l(Fq)[X]/(X^m-1). This permits to construct new classes of codes, namely quasi-BCH and quasi-evaluation codes. We study the parameters of such codes and propose a decoding algorithm up to half the designed minimum distance. We even found one new quasi-cyclic code with better parameters than known [189, 11, 125]_F4 and 48 derivated codes beating the known bounds as well.
(18/08/2011)
FOS: Computer and information sciences, cyclic codes, Matrices over special rings (quaternions, finite fields, etc.), Decoding, Computer Science - Information Theory, [INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS], [INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS], key equation, quasi-cyclic codes, Matrix rings, [MATH.MATH-IT] Mathematics [math]/Information Theory [math.IT], Key equation, left ideals, evaluation codes, matrix rings, decoding algorithm, Information Theory (cs.IT), [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT], Left ideals, principal ideals, 004, Quasi-cyclic codes, [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT], Evaluation codes, [INFO.INFO-IT] Computer Science [cs]/Information Theory [cs.IT], Cyclic codes
FOS: Computer and information sciences, cyclic codes, Matrices over special rings (quaternions, finite fields, etc.), Decoding, Computer Science - Information Theory, [INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS], [INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS], key equation, quasi-cyclic codes, Matrix rings, [MATH.MATH-IT] Mathematics [math]/Information Theory [math.IT], Key equation, left ideals, evaluation codes, matrix rings, decoding algorithm, Information Theory (cs.IT), [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT], Left ideals, principal ideals, 004, Quasi-cyclic codes, [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT], Evaluation codes, [INFO.INFO-IT] Computer Science [cs]/Information Theory [cs.IT], Cyclic codes
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