On Hull-Variation Problem of Equivalent Linear Codes
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The intersection C⋂ C^⊥ (C⋂ C^⊥_h) of a linear code C and its Euclidean dual C^⊥ (Hermitian dual C^⊥_h) is called the Euclidean (Hermitian) hull of this code. The construction of an entanglement-assisted quantum code from a linear code over F_q or F_q^2 depends essentially on the Euclidean hull or the Hermitian hull of this code. Therefore it is natural to consider the hull-variation problem when a linear code C is transformed to an equivalent code v· 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. A general method to construct hull-decreasing or hull-increasing equivalent linear codes is proposed. We prove that for a nonnegative integer h satisfying 0 ≤ h ≤ 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 F_2^s satisfying d≥ 2 and d^⊥≥ 2 is equivalent to a linear one-dimension hull code under a weak condition. Several new families of negacyclic LCD codes and BCH LCD codes over 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 EAQEC codes including MDS and almost MDS entanglement-assisted quantum codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.
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