Classification of optimal additive toeplitz codes over GF(4)

Abstract

We denote GF_4={0,1,w,w^2} where w^2=w+1. An additive code C over GF_4 of length n is an additive subgroup of GF_4^n . C contains 2^k codewords for some 0<=k<= 2n, and can be defined by a kxn generator matrix, with entries from GF_4, whose rows span C additively. An additive code with minimum distance d is called an (n,2^k,d), code. If a code has highest possible minimum distance, denoted by max d_max, it is called optimal. We say that two additive codes C_1 and C_2 over GF_4 are equivalent provided there is a map sending the codewords of C_1 onto the codewords of C_2 where the map consists of a permutation of coordinates (or columns of the generator matrix), followed by a scaling of coordinates by nonzero elements of GF_4, followed by conjugation of some of the coordinates. The conjugation of x in GF_4 is defined by x^-=x^2 . For a code of length n , there is a total of 6^nn!. Gaborit et al. [3] determined the equivalence or inequivalence of two additive codes over GF_4 by an algorithm. In this study, we introduce additive toeplitz codes over GF_4. The additive toeplitz codes are a generalization of additive circulant codes over GF_4. We provide some theorems to partially classify optimal additive toeplitz codes (OATC). Then, we give a new algorithm that fully classifies OATC by combining these theorems with Gaborit’s algorithm. We classify OATC over GF_4 of length up to 13 , except for length 9 . We obtain 2 inequivalent optimal additive toeplitz codes (IOATC) that are non-circulant codes of length 5 , 92 of length 8 , and 39 of length 11. Moreover, we construct 49 IOATC that are non-circulant codes of length 9 .