Shortest self-orthogonal embeddings of binary linear codes

Abstract

There has been recent interest in the study of shortest self-orthogonal embeddings of binary linear codes, since many such codes are optimal self-orthogonal codes. Several authors have studied the length of a shortest self-orthogonal embedding of a given binary code $\mathcal{C}$, or equivalently, the minimum number of columns that must be added to a generator matrix of $\mathcal{C}$ to form a generator matrix of a self-orthogonal code. In this paper, we use properties of the hull of a linear code to determine the length of a shortest self-orthogonal embedding of any binary linear code. We focus on the examples of Hamming codes and Reed-Muller codes. We show that a shortest self-orthogonal embedding of a binary Hamming code is self-dual, and propose two algorithms to construct self-dual codes from Hamming codes ${\mathcal{H}}_r$. Using these algorithms, we construct a self-dual $[22,11,6]$ code, called the shortened Golay code, from the binary $[15,11,3]$ Hamming code ${\mathcal{H}}_4$, and construct a self-dual $[52,26,8]$ code from the binary $[31,26,3]$ Hamming code ${\mathcal{H}}_5$. We use shortest SO embeddings of linear codes to obtain many inequivalent optimal self-orthogonal codes of dimension $7$ and $8$ for several lengths. Four of the codes of dimension $8$ that we construct are codes with new parameters such as $[91,8,42],[98,8,46],[114,8,54]$, and $[191,8,94]$.

17 pages