On the Hull-Variation Problem of Equivalent Linear Codes

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