A Note on a Subcode of a Linear q - Ary Code of Length N and an Algorithm for Calculating Minimum Distance

Dr. M. Mary Jansi Rani, M. Manikandan

Abstract: This paper deals with the calculation of minimum distance of a q-ary linear code of length n. The set of code words having the left most coordinate position O forms a subcode. If C is an [n, k, d] code, the subcode Co so considered is of dimensions k-1. The coset leaders in C/Co give the method calculating minimum distance. The method does not make use of the known techniques using a parity check matrix H. In phillipe Delsarte Four fundametal parameters of a code and their combinatorial significance, information and control 23, 407 438 (1973), the inner product of two vectors a and b in is considered using group characters of a finite abelian group (F, +) of order q over, The cyclotomic field of complex Vth roots of unity. The dual code is defined via the inner product of a, b n, If reduces to the classical concept for linear codes over finite fields. However, if a = a0 a1. . . . an, b = b0 b1. . . bn-1. The inner product a, b = could be interpreted using a cyclotomic extension of Fq via trace of an element in. This give get another interpretation of the inner product a. b.

Keywords: Cyclotomic cosets, minimum Distance, Co ordinate Position, Coset Leader, Subcode, Inner Product