Shortest self-orthogonal embeddings of binary linear codes
Junmin An, Nathan Kaplan, Jon-Lark Kim, Jinquan Luo, Guodong Wang
TL;DR
This work introduces a hull-based framework to determine the minimal length of shortest self-orthogonal embeddings for binary linear codes, yielding precise formulas: the length is $n+k-\ell$ when $k-\ell$ is odd, and either $n+k-\ell$ or $n+k-\ell+1$ when $k-\ell$ is even, with the exact option depending on whether the code is even or odd. It analyzes Reed–Muller and binary Hamming codes, proving that Hamming code embeddings are self-dual and supplying algorithms to construct self-dual codes from Hamming codes; notably, it obtains a $[52,26,8]$ code from $\mathcal{H}_5$ and a shortened Golay $[22,11,6]$ code from $\mathcal{H}_4$. The paper also provides an exhaustive and an orthogonal-matrix-based classification of shortest SO embeddings of Hamming codes, illustrating deep connections between Hamming codes and Golay-type constructions. Leveraging these embeddings, it produces numerous optimal self-orthogonal codes of dimensions $7$ and $8$, including several with new parameters such as $[91,8,42],\ [98,8,46],\ [114,8,54],\ [191,8,94]$, thereby expanding the catalog of high-distance SO codes and informing applications in quantum coding and design theory.
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 $[191, 8, 94]$.