Rank and Kernel of F_p-Additive Generalised Hadamard Codes
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A subset of a vector space F_q^n is K-additive if it is a linear space over the subfield K⊆F_q. Let q=p^e, p prime, and e>1. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are F_p-additive are established. For specific ranks and dimensions of the kernel within these bounds, F_p-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 F_p-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.
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