Linear Hadamard Codes Research Articles

On $\mathbb{Z}_{\text{8}}$ -Linear Hadamard Codes: Rank and Classification

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 either the rank or the dimension of the kernel can be used to give a complete classification for the $\mathbb {Z}_{4}$ -linear Hadamard codes. However, when $s > 2$ , the dimension of the kernel of $\mathbb {Z}_{2^{s}}$ -linear Hadamard codes of length $2^{t}$ only provides a complete classification for some values of $t$ and $s$ . In this paper, the rank of these codes is computed for $s=3$ . Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once $t\geq 3$ is fixed. In this case, the number of nonequivalent such codes is also established.

Partial Permutation Decoding for Several Families of Linear and <inline-formula> <tex-math notation="LaTeX">${\mathbb{Z}}_4$ </tex-math> </inline-formula>-Linear Codes

A general criterion to obtain $s$ -PD-sets of minimum size $s+1$ for partial permutation decoding, which enable correction up to $s$ errors, for systematic codes over a finite field ${ {F}}_{q}$ and ${ {Z}}_{4}$ -linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over ${ {F}}_{q}$ , ${ {Z}}_{4}$ -linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new $s$ -PD-sets of size $s+1$ for ${ {Z}}_{4}$ -linear Hadamard codes of type $2^\gamma 4^\delta $ , for all $\delta \geq 4$ and $1 ; and for ${ {Z}}_{4}$ -linear simplex codes of type $4^{m}$ , for all $m\geq 2$ and $1 , are also provided.

A note on regular subgroups of the automorphism group of the linear Hadamard code

A note on regular subgroups of the automorphism group of the linear Hadamard code

On $$\mathbb {Z}_{2^s}$$ Z 2 s -linear Hadamard codes: kernel and partial classification

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 of length $$2^t$$ and its dimension are established for $$s > 2$$ . Moreover, we prove that this invariant only provides a complete classification for some values of t and s. The exact amount of nonequivalent such codes are given up to $$t=11$$ for any $$s\ge 2$$ , by using also the rank.

On two-weight $$\mathbb {Z}_{2^k}$$ Z 2 k -codes

We determine the possible homogeneous weights of regular projective two-weight codes over $$\mathbb {Z}_{2^k}$$ of length $$n>3$$ , with dual Krotov distance $$d^{\lozenge }$$ at least four. The determination of the weights is based on parameter restrictions for strongly regular graphs applied to the coset graph of the dual code. When $$k=2$$ , we characterize the parameters of such codes as those of the inverse Gray images of $$\mathbb {Z}_4$$ -linear Hadamard codes, which have been characterized by their types by several authors.

Partial permutation decoding for binary linear and $$Z_4$$ Z 4 -linear Hadamard codes

In this paper, s- $${\text {PD}}$$ -sets of minimum size $$s+1$$ for partial permutation decoding for the binary linear Hadamard code $$H_m$$ of length $$2^m$$ , for all $$m\ge 4$$ and $$2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1$$ , are constructed. Moreover, recursive constructions to obtain s- $${\text {PD}}$$ -sets of size $$l\ge s+1$$ for $$H_{m+1}$$ of length $$2^{m+1}$$ , from an s- $${\text {PD}}$$ -set of the same size for $$H_m$$ , are also described. These results are generalized to find s- $${\text {PD}}$$ -sets for the $${\mathbb {Z}}_4$$ -linear Hadamard codes $$H_{\gamma , \delta }$$ of length $$2^m$$ , $$m=\gamma +2\delta -1$$ , which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type $$2^\gamma 4^\delta $$ . Specifically, s-PD-sets of minimum size $$s+1$$ for $$H_{\gamma , \delta }$$ , for all $$\delta \ge 3$$ and $$2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1$$ , are constructed and recursive constructions are described.

Classification of the <inline-formula> <tex-math notation="LaTeX">$Z_{2}Z_{4}$ </tex-math></inline-formula>-Linear Hadamard Codes and Their Automorphism Groups

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 exactly $[(t-1)/2]$ and $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$, with $\alpha=0$ and $\alpha\not=0$, respectively, for all $t\geq 3$. In this paper, it is shown that each $Z_2Z_4$-linear Hadamard code with $\alpha=0$ is equivalent to a $Z_2Z_4$-linear Hadamard code with $\alpha\not=0$; so there are only $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$. Moreover, the order of the monomial automorphism group for the $Z_2Z_4$-additive Hadamard codes and the permutation automorphism group of the corresponding $Z_2Z_4$-linear Hadamard codes are given.

Characterization of the automorphism group of quaternary linear Hadamard codes

A quaternary linear Hadamard code $${\mathcal{C}}$$ is a code over $${\mathbb{Z}_4}$$ such that, under the Gray map, gives a binary Hadamard code. The permutation automorphism group of a quaternary linear code $${\mathcal{C}}$$ of length n is defined as $${{\rm PAut}(\mathcal{C}) = \{\sigma \in S_{n} : \sigma(\mathcal{C}) = \mathcal{C}\}}$$ . In this paper, the order of the permutation automorphism group of a family of quaternary linear Hadamard codes is established. Moreover, these groups are completely characterized by computing the orbits of the action of $${{\rm PAut}(\mathcal{C})}$$ on $${\mathcal{C}}$$ and by giving the generators of the group. Since the dual of a Hadamard code is an extended 1-perfect code in the quaternary sense, the permutation automorphism group of these codes is also computed.

A class of nonorthogonal rate-one space-time block codes with controlled interference

Complex, rate one, orthogonal block space-time codes (STCs) exist only for two transmit antennas and were proposed by Alamouti (1998). Designing rate-one block STCs for more than two transmit antennas implies giving up orthogonality. For more than two transmit antennas a class of rate one block STCs with a controlled number of interference terms per detected symbol is presented. Conditions that link the matrix formulation of the receive equation to the matrix formulation of a multiuser detection scheme are discussed. The block STCs design problem is closely related to the class of linear real Hadamard codes.