2020/01/30 by Dougherty, Steven T., Rifà, Josep, Villanueva, Mercè
#FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2001.11609
A subset of a vector space \mathbbFqn is K-additive if it is a linear space over the subfield K⊆ \mathbbFq. Let q=pe, p prime, and e>1. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are \mathbbFp-additive are established. For specific ranks and dimensions of the kernel within these bounds, \mathbbFp-additive GH codes are constructed. Moreover, for the case e=2, it is shown that the given bounds are tight and it is possible to construct an \mathbbFp-additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.