Abstract
We prove that if n >k^2 then a k-dimensional linear code of length n over {mathbb F}_{q^2} has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code. In the contrary case we prove that truncations of linear codes to codes equivalent to Hermitian self-orthogonal linear codes occur when the columns of a generator matrix of the code do not impose independent conditions on the space of Hermitian forms. In the case that there are more than n common zeros to the set of Hermitian forms which are zero on the columns of a generator matrix of the code, the additional zeros give the extension of the code to a code that has a truncation which is equivalent to a Hermitian self-orthogonal code.
Highlights
The main motivation to study Hermitian self-orthogonal codes is their application to quantum error-correcting codes
Theorem 9 implies that the set of points X imposes n conditions on the space of Hermitian forms if and only if dim P (C) = 0 which, by Theorem 2, is if and only if no truncation of C is equivalent to a Hermitian self-orthogonal code
Theorem 14 indicates that to extend a linear code C to a Hermitian selforthogonal code, we should calculate the set of common zeros of the Hermitian forms which are zero on the columns of a generator matrix for C
Summary
The main motivation to study Hermitian self-orthogonal codes is their application to quantum error-correcting codes. The most prevalent and applicative quantum codes are qubit codes, in which the quantum state is encoded on n quantum particles with two-states. In this case, the quantum code is a subspace of (C2)⊗n. A quantum code is a subspace of (Cq)⊗n. A qubit is referred to as a quqit. A quantum code with minimum distance d is able to detect errors, which act non-trivially on the code space, on up to d − 1 of the quqits and correct errors on up to
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