Constructions of Nonequivalent F p -Additive Generalised Hadamard Codes
Steven T. Dougherty, Josep Rifà, Mercè Villanueva · 2020
A subset of a vector space Fqnis K-additive if it is a linear space over the subfield K ⊆ Fq. Let q = pe, p prime, and e > 1. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are Fp-additive are established. For specific ranks and dimensions of the kernel within these bounds, Fp-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 Fp-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.