Related papers: Equivalences among Z_{p^s}-linear Generalized Hada…
The $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_p^{\alpha_1} \times \mathbb{Z}_{p^2}^{\alpha_2} \times \cdots \times \mathbb{Z}_{p^s}^{\alpha_s}$, and can be seen as linear codes over…
The $\Z_p\Z_{p^2}$-additive codes are subgroups of $\Z_p^{\alpha_1} \times \Z_{p^2}^{\alpha_2}$, and can be seen as linear codes over $\Z_p$ when $\alpha_2=0$, $\Z_{p^2}$-additive codes when $\alpha_1=0$, or $\Z_2\Z_4$-additive codes when…
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…
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)…
The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3}$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard…
The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3}$, and can be seen as linear codes over $\mathbb{Z}_2$ when…
A $Z_2Z_4$-linear Hadamard code of length $\alpha+2\beta=2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2Z_4$-additive code with $\alpha$ binary coordinates and $\beta$ quaternary coordinates. It is known that there are…
A code $C$ is called $\Z_p\Z_{p^2}$-linear if it is the Gray image of a $\Z_p\Z_{p^2}$-additive code, where $p>2$ is prime. In this paper, the rank and the dimension of the kernel of $\Z_p\Z_{p^2}$-linear codes are studied. Two bounds of…
The $\Z_{2^s}$-additive codes are subgroups of $\Z^n_{2^s}$, and can be seen as a generalization of linear codes over $\Z_2$ and $\Z_4$. A $\Z_{2^s}$-linear code is a binary code which is the Gray map image of a $\Z_{2^s}$-additive code. We…
The ranks and kernels of generalized Hadamard matrices are studied. It is proven that any generalized Hadamard matrix $H(q,\lambda)$ over $F_q$, $q>3$, or $q=3$ and $\gcd(3,\lambda)\not =1$, generates a self-orthogonal code. This result…
Codes from generalized Hadamard matrices have already been introduced. Here we deal with these codes when the generalized Hadamard matrices are cocyclic. As a consequence, a new class of codes that we call generalized Hadamard full…
A code $C$ is called $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear if it is the Gray image of a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code. For any prime number $p$ larger than $3$, the bounds of the rank of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear…
All generalized Hadamard matrices of order 18 over a group of order 3, H(6,3), are enumerated in two different ways: once, as class regular symmetric (6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of…
A new generalization of the Gray map is introduced. The new generalization $\Phi: Z_{2^k}^n \to Z_{2}^{2^{k-1}n}$ is connected with the known generalized Gray map $\phi$ in the following way: if we take two dual linear $Z_{2^k}$-codes and…
Hadamard full propelinear codes (HFP-codes) are introduced and their equivalence with Hadamard groups is proven (on the other hand, it is already known the equivalence of Hadamard groups with relative $(4n,2,4n,2n)$-difference sets in a…
Generalized Goppa codes are defined by a code locator set $\mathcal{L}$ of polynomials and a Goppa polynomial $G(x)$. When the degree of all code locator polynomials in $\mathcal{L}$ is one, generalized Goppa codes are classical Goppa…
A linear code of length $n$ over a finite chain ring $R$ with residue field $\F_q$ is a $R$-submodule of $R^n$. A $R$-linear code is a code over $\F_q$ (not necessarily linear) which is the generalized Gray map image of a linear code over…
Partitions of the binary linear Hamming code into Preparata-like codes are known to induce line-parallelisms of PG$(n, 2)$. In this paper, we show that if $P$ is any Preparata-like code contained in the binary linear Hamming code $H$ of the…
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…
Linear codes over finite rings become one of hot topics in coding theory after Hommons et al.([4], 1994) discovered that several remarkable nonlinear binary codes with some linear-like properties are the images of Gray map of linear codes…