The geometry of Hermitian self-orthogonal codes

2021/08/18 by Ball, Simeon, Vilar, Ricard
#11E39 #51Exx #94Bxx #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Information Theory (cs.IT) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2108.08088

Abstract

We prove that if n >k² then a k-dimensional linear code of length n over \mathbb F_q² 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.

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