The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1} \times \mathbb{Z}_4^{α_2} \times \mathbb{Z}_8^{α_3}$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which is the Gray map image of a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes of type $(α_1,α_2, α_3;t_1,t_2, t_3)$ with $α_1 \neq 0$, $α_2 \neq 0$, $α_3 \neq 0$, $t_1\geq 1$, $t_2 \geq 0$, and $t_3\geq 1$ is known. In this paper, we generalize some known results for $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard codes to $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes with $α_1 \neq 0$, $α_2 \neq 0$, and $α_3 \neq 0$. First, we show for which types the corresponding $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes of length $2^t$ are nonlinear. For these codes, we compute the kernel and its dimension, which allows us to give a partial classification of these codes. Moreover, for $3 \leq t \leq 11$, we give a complete classification by providing the exact amount of nonequivalent such codes. We also prove the existence of several families of infinite such nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, which are not equivalent to any other constructed $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code, nor to any $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code, nor to any previously constructed $\mathbb{Z}_{2^s}$-linear Hadamard code with $s\geq 2$, with the same length $2^t$.
The $\Z_p\Z_{p^2}$-additive codes are subgroups of $\Z_p^{α_1} \times \Z_{p^2}^{α_2}$, and can be seen as linear codes over $\Z_p$ when $α_2=0$, $\Z_{p^2}$-additive codes when $α_1=0$, or $\Z_2\Z_4$-additive codes when $p=2$. A $\Z_p\Z_{p^2}$-linear generalized Hadamard (GH) code is a GH code over $\Z_p$ which is the Gray map image of a $\Z_p\Z_{p^2}$-additive code. In this paper, we generalize some known results for $\Z_p\Z_{p^2}$-linear GH codes with $p=2$ to any $p\geq 3$ prime when $α_1 \neq 0$. First, we give a recursive construction of $\Z_p\Z_{p^2}$-additive GH codes of type $(α_1,α_2;t…
The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1} \times \mathbb{Z}_4^{α_2} \times \mathbb{Z}_8^{α_3}$, and can be seen as linear codes over $\mathbb{Z}_2$ when $α_2=α_3=0$, $\mathbb{Z}_4$-additive or $\mathbb{Z}_8$-additive codes when $α_1=α_3=0$ or $α_1=α_2=0$, respectively, or $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $α_3=0$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which is the Gray map image of a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive code. In this paper, we generalize some known results fo…
The $\mathbb{Z}_{2^s}$-additive codes are subgroups of $\mathbb{Z}^n_{2^s}$, and can be seen as a generalization of linear codes over $\mathbb{Z}_2$ and $\mathbb{Z}_4$. A $\mathbb{Z}_{2^s}$-linear Hadamard code is a binary Hadamard code which is the Gray map image of a $\mathbb{Z}_{2^s}$-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the $\mathbb{Z}_4$-linear Hadamard codes. In this paper, the kernel of $\mathbb{Z}_{2^s}$-linear Hadamard codes and its dimension are established for $s > 2$. Moreover, we prove that this invariant only…
The kernel-block rank profiles of the $\mathbb{Z}_2\mathbb{Z}_4$-linear and the $\mathbb{Z}_{2^s}$-linear Hadamard codes, and a complete classification of the $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes
A subset of a vector space $\mathbb{F}_q^n$ is $K$-additive if it is a linear space over the subfield $K\subseteq \mathbb{F}_q$. 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 $\mathbb{F}_p$-additive are established. For specific ranks and dimensions of the kernel within these bounds, $\mathbb{F}_p$-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 $\mathbb{F}_p$-additive GH code for all allowable ranks and dimensions of t…
In this paper, we study lattice coding for Rician fading wireless channels. This is motivated in particular by preliminary studies suggesting the Rician fading model for millimeter-wavelength wireless communications. We restrict to lattice codes arising from rotations of $\mathbb{Z}^n$, and to a single-input single-output (SISO) channel. We observe that several lattice design criteria suggest the optimality of Hadamard rotations. For instance, we prove that Hadamard rotations maximize the diamond-packing density among all rotated $\mathbb{Z}^n$ lattices. Finally, we provide simulations to show…
The $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_p^{α_1} \times \mathbb{Z}_{p^2}^{α_2} \times \cdots \times \mathbb{Z}_{p^s}^{α_s}$, and can be seen as linear codes over $\mathbb{Z}_p$ when $α_i=0$ for all $i \in \{2,\dots, s\}$, a $\mathbb{Z}_{p^s}$-additive code when $α_i=0$ for all $i \in \{1,\dots, s-1\}$ , or a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code when $s=2$, or $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $p=2$ and $s=2$. A $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear generalized Hadamard (GH) code is a GH code ov…
In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of Ó Catháin and Swartz. That is, we show how, if given a Butson Hadamard matrix over the $k^{\rm th}$ roots of unity, we can construct a larger Butson matrix over the $\ell^{\rm th}$ roots of unity for any $\ell$ dividing $k$, provided that any prime $p$ dividing $k$ also d…
Permutation decoding is a technique which involves finding a subset $S$, called PD-set, of the permutation automorphism group of a code $C$ in order to assist in decoding. An explicit construction of $\left \lfloor{\frac{2^m-m-1}{1+m}} \right \rfloor$-PD-sets of minimum size $\left \lfloor{\frac{2^m-m-1}{1+m}} \right \rfloor + 1$ for partial permutation decoding for binary linear Hadamard codes $H_m$ of length $2^m$, for all $m\geq 4$, is described. Moreover, a recursive construction to obtain $s$-PD-sets of size $l$ for $H_{m+1}$ of length $2^{m+1}$, from a given $s$-PD-set of the same size f…
In this work, we study functions that can be obtained by restricting a vectorial Boolean function $F \colon \mathbb{F}_2^n \rightarrow \mathbb{F}_2^n$ to an affine hyperplane of dimension $n-1$ and then projecting the output to an $n-1$-dimensional space. We show that a multiset of $2 \cdot (2^n-1)^2$ EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on $\mathbb{F}_2^n$. Further, for all of the known quadratic APN functions in dimension $n < 10$, we determine the restrictions that are also APN. Moreover, we construct 6,368 new quadratic APN fun…
In the paper, we firstly study the algebraic structures of $\mathbb{Z}_p \mathbb{Z}_{p^k}$-additive cyclic codes and give the generator polynomials and the minimal spanning set of these codes. Secondly, a necessary and sufficient condition for a class of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive codes whose Gray images are linear (not necessarily cyclic) over $\mathbb{Z}_p$ is given. Moreover, as for some special families of cyclic codes over $\mathbb{Z}_{9}$ and $\mathbb{Z}_3 \mathbb{Z}_{9}$, the linearity of the Gray images is determined.
