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On non-full-rank perfect codes over finite fields (1704.02627v1)
Published 9 Apr 2017 in cs.IT and math.IT
Abstract: 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 $n = (q{m}-1)/(q-1)$ is a $q$-ary constant-weight code with Hamming weight equals to $q{m - 1}$ 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.