On the Hull-Variation Problem of Equivalent Linear Codes
Hao Chen · IEEE Transactions on Information Theory · 2023
The intersection${\mathbf{C}}\bigcap {\mathbf{C}}^{\perp }$(${\mathbf{C}}\bigcap {\mathbf{C}}^{\perp _{h}}$) of a linear code${\mathbf{C}}$and its Euclidean dual${\mathbf{C}}^{\perp }$(Hermitian dual${\mathbf{C}}^{\perp _{h}}$) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code${\mathbf{C}}$is transformed to an equivalent code${\mathbf{v}} \cdot {\mathbf{C}}$. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We prove that for a nonnegative integer$h$satisfying$0 \leq h \leq n-1$, a linear$[2n], [n]_{q}$self-dual code is equivalent to a linear$h$-dimension hull code. On the opposite direction we prove that a linear LCD code over${\mathbf{F}}_{2^{s}}$satisfying$d\geq 2$and$d^{\perp } \geq 2$is equivalent to a linear one-dimension hull code under a weak condition. Several new families of LCD negacyclic codes and LCD BCH codes over${\mathbf{F}}_{3}$are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new entanglement-assisted quantum error-correction (EAQEC) codes including MDS and almost MDS EAQEC codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.