Rank and Kernel of Additive Generalized Hadamard Codes

A subset of a vector space
$\mathbb {F}_{q}^{n}$
is additive if it is a linear space over the field
$\mathbb {F}_{p}$
, where
$q=p^{e}$
,
$p$
prime, and
$e>1$
. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, 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 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.