Embedding Linear Codes Into Self-Orthogonal Codes and Their Optimal Minimum Distances

We obtain a characterization on self-orthogonality for given binary linear code in terms of the number column vectors its generator matrix, which extends result Bouyukliev et al. (2006). As an application, we give algorithmic method to embed k-dimensional C ( k = 3,4) into self-orthogonal shortest length has same dimension and minimum distance d' ? d( ). For > 4, suggest recursive code. also new explicit formulas distances optimal codes any n with 4 \not ? 6, 13,14,21,22,28,29 (mod 31) 5.