Construction of binary self-orthogonal codes (Q6542670)

A family \(\Delta\subseteq \mathbb{F}_2^m\) is called a simplicial complex if \(u\in\Delta\) and \(v\in \mathbb{F}_2^m\) with \(\operatorname{supp}(v) \subseteq \operatorname{supp}(u)\) imply \(v\in\Delta.\) When \(D = \{g_1, g_2,\ldots, g_n\} \subseteq \mathbb{F}_2^m,\) let \(G\) be the \(m\times n\) matrix \(G = [g_1^T g_2^T \ldots g_n^T],\) then the following linear code of length \(n\) over \(\mathbb{F}_2\) can be obtained: \(C_D = \{c_u = (u\cdot g_1, u \cdot g_2, \ldots, u \cdot g_n) : u \in\mathbb{F}_2^m\},\) and the set \(D\) is said to be the defining set of the code \(C_D.\)\N\NIn this work, the authors, using simplicial complexes, propose two new construction methods for binary self-orthogonal codes using previously known self-orthogonal codes. This is achieved by considering a change of the generator matrix of a binary linear code. The method starts with a self-orthogonal code having defining set \(D_1.\) With the defining set \(D = \{(d_1, d_2) : d_1\in D_1, d_2\in D_2\} \subseteq \mathbb{F}_2^{t_1+t_2},\) where \(D_1\subseteq \mathbb{F}_2^{t_1}\) and \(D_2 \subseteq \mathbb{F}_2^{t_2}\) a change in the generator matrix is achieved such that a new binary linear code is obtained. A special defining set \(D_2\) is found such that the code with defining set \(D\) is still self-orthogonal.\N\NUsing these two new constructions, four infinite classes of binary self-orthogonal codes are shown. A determination of their weight distributions and the minimum distances of their dual codes is performed. As a result, a class of optimal linear codes and a class of almost optimal linear codes with respect to the sphere packing bound are found. Sufficient examples show that the proposed strategy is effective.