2017/04/09 by Alexander M. Romanov, Romanov, Alexander M.
Computer Science · Mathematics · #94B25 #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #math.IT #msc:94B25
paper · pdf · doi:10.48550/arxiv.1704.02627
arxiv created 2017/04/09 · arxiv updated 2017/04/11
The paper deals with the perfect 1-error correcting codes over a finite field with q elements (briefly q-ary 1-perfect codes). We show that the orthogonal code to the q-ary non-full-rank 1-perfect code of length n = (qm-1)/(q-1) is a q-ary constant-weight code with Hamming weight equals to qm - 1 where m is any natural number not less than two. We derive necessary and sufficient conditions for q-ary 1-perfect codes of non-full rank. We suggest a generalization of the concatenation construction to the q-ary case and construct the ternary 1-perfect codes of length 13 and rank 12.