On some binary symplectic self-orthogonal codes (Q2140838)

Inspired by previous works on Euclidean and Hermitian self-dual code this paper studies symplectic self-orthogonal codes over finite fields. The symplectic inner product is used in quantum error-correcting codes. Denoting by \(F_q\) the finite field of order \(q\) the symplectic inner product is defined as \(\langle x, y\rangle_S= x\Omega y^T = \langle u, v'\rangle_E - \langle u', v\rangle _E,\) where \(\Omega=\left( \begin{array}{cc} 0 & I_n \\ -I_n & 0 \\ \end{array} \right)\) and \(x=(u|v), y=(u'|v')\in F_q^{2n}.\) Thus a symplectic weight and distance are defined as follows: \(\mathrm{wt}_S(u|v)=|\{i: (u,v)\neq(0,0)\}|\), \(\mathrm{d}_S(x,y)=\mathrm{wt}_S(x-y)\), respectively. For an \(F_q\)-linear code \(C\) in \(F_q^{2n}\), the symplectic dual code is defined as \(C^{\bot_S}=\{x\in F_q^{2n} : \langle x, c\rangle_S = 0\) for all \(c \in C\}\) and a code \(C\) is symplectic self-dual if \(C\) coincides with itself and is self-orthogonal if \(C\subseteq C^{\bot_S}\). Some characterizations of symplectic self-orthogonal codes are given using a generator matrix \(G\) satisfying the condition \(G\Omega G^T = 0.\) Furthermore, the authors provide a necessary and sufficient condition for determining symplectic self-orthogonal codes and present some new constructions of symplectic self-orthogonal codes. By applying these constructions, some new classes of binary symplectic self-orthogonal codes are obtained and their symplectic weight distribution is explicitly established.