Binary Hadamard Code Research Articles

Butson full propelinear codes

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{mathrm{th}} roots of unity, we can construct a larger Butson matrix over the ell mathrm{th} roots of unity for any ell dividing k, provided that any prime p dividing k also divides ell . We prove that a {mathbb {Z}}_{p^s}-additive code with p a prime number is isomorphic as a group to a BH-code over {mathbb {Z}}_{p^s} and the image of this BH-code under the Gray map is a BH-code over {mathbb {Z}}_p (binary Hadamard code for p=2). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided.

Construction of binary Hadamard codes and their s-PD sets

This article presents the construction of binary linear Hadamard codes with parameters (2n,4n,n) over the fields F4 and F8. We generalize this construction to the field $F_{2^{k}}$ to generate Hadamard codes with parameters $\left (2^{k-1}n, 2^{k}n, 2^{k-2}n\right )$ for $k\in \mathbb {N}$ . We construct s-PD sets of size s + 1, a special subset of the permutation automorphism group of a code, which enables the correction of s errors for Hadamard codes over the field F4. We also discuss the decoding algorithm for these Hadamard codes.

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.

Equivalences among [formula omitted]-linear Hadamard codes

The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z2s-linear Hadamard codes of length 2t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t=11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s∈{2,3}, the full classification of the Z2s-linear Hadamard codes of length 2t is established by giving the exact number of such codes.

On the Rank of [formula omitted]-linear Hadamard Codes

The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-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 Z4-linear Hadamard codes. However, when s>2, the dimension of the kernel of Z2s-linear Hadamard codes of length 2t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is given for s=3. Moreover, it is shown that this invariant, along with the dimension of the kernel, provides a complete classification, once t≥3 is fixed. In this case, the number of nonequivalent such codes is also established.

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.

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.

Construction and classification of [formula omitted]-linear Hadamard codes

The Z2s-additive and Z2Z4-additive codes are subgroups of Z2sn and Z2α×Z4β, respectively. Both families can be seen as generalizations of linear codes over Z2 and Z4. A Z2s-linear (resp. Z2Z4-linear) Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive (resp. Z2Z4-additive) code. It is known that there are exactly ⌊t−12⌋ and ⌊t2⌋ nonequivalent Z2Z4-linear Hadamard codes of length 2t, with α=0 and α≠0, respectively, for all t≥3. In this paper, new Z2s-linear Hadamard codes are constructed for s>2, which are not equivalent to any Z2Z4-linear Hadamard code. Moreover, for each s>2, it is claimed that the new constructed nonlinear Z2s-linear Hadamard codes of length 2t are pairwise nonequivalent.

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.

Multicode MIMO Systems With Quaternary LCZ and ZCZ Sequences

In this paper, we propose multicode multiple-input-multiple-output (MIMO) systems with quaternary low-correlation zone (LCZ) and zero-correlation zone (ZCZ) sequences as spreading codes. Quaternary LCZ and ZCZ sequences have very low correlation values when the time shifts between these sequences are within the predetermined correlation zone, and thus, the multiuser or multipath interference can be substantially reduced when the delay is within a few chips. The bit error probability of the proposed systems is theoretically analyzed, which is numerically confirmed. It is also numerically shown that the performance of the multicode MIMO systems with quaternary LCZ and ZCZ sequences is better than that of the conventional multicode MIMO systems with quaternary spreading codes constructed from pairs of binary Hadamard codes.

Rank and Kernel of Binary Hadamard Codes

In this paper, the rank and the dimension of the kernel for (binary) Hadamard codes of length a power of two are studied. In general, it is well-known that the rank of a Hadamard code of length n=2/sup t/ is a value in {t+1,...,n/2}. In the present paper, the range of possible values for the dimension of the kernel is computed and a construction of Hadamard codes of length n=2/sup t/ for each one of these values is given. Lower and upper bounds for the rank and dimension of the kernel of a Hadamard code of length n=2/sup t/ are also established. Finally, we construct Hadamard codes for all possible ranks and dimension of kernels between these bounds.