Construction Of Self-orthogonal Codes Research Articles

Constructions of Self-Orthogonal Codes From Hulls of BCH Codes and Their Parameters

Self-orthogonal codes are an interesting type of linear codes due to their wide applications in communication and cryptography. It is known that self-orthogonal codes are often used to construct quantum error-correcting codes, which can protect quantum information in quantum computations and quantum communications. Let C be an [n, k] cyclic code over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , where F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> is the finite field of order q. The hull of C is defined to be the intersection of the code and its dual. In this paper, we will employ the defining sets of cyclic codes to present two general characterizations of the hulls that have dimension k - 1 or k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> - 1, where k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> is the dimension of the dual code C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> . Several sufficient and necessary conditions for primitive and projective BCH codes to have (k - 1)-dimensional (or (k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> -1)dimensional) hulls are also developed by presenting lower and upper bounds on their designed distances. Furthermore, several classes of self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. The dimensions and minimum distances of some self-orthogonal codes are determined explicitly. In addition, several optimal codes are obtained.

Self-orthogonal codes constructed from orbit matrices of 2-designs and quotient matrices of divisible designs

Abstract In 2003 Harada and Tonchev showed a construction of self-orthogonal codes from orbit matrices of block designs with fixed-point-free automorphisms. We describe a construction of self-orthogonal codes from orbit matrices of 2-designs admitting certain automorphisms with fixed points (and blocks). Further, we present a construction of self-orthogonal codes from quotient matrices of divisible designs and divisible design graphs.

Construction of self-orthogonal codes from combinatorial designs

Self-orthogonal codes are constructed from matrices generated according to parameters of combinatorial designs. An approach towards generating designs and such matrices is considered. Some classification results on self-orthogonal codes are also presented.