Weighing matrices and spherical codes

Mutually unbiased weighing matrices (MUWM) are closely related to an antipodal spherical code with 4 angles. In the present paper, we clarify the relationship between MUWM and the spherical sets, and give the complete solution about the maximum size of a set of MUWM of weight 4 for any order. Moreover we describe some natural generalization of a set of MUWM from the viewpoint of spherical codes, and determine several maximum sizes of the generalized sets. They include an affirmative answer of the problem of Best, Kharaghani, and Ramp.Comment: Title is changed from "Association schemes related to weighing matrices

Classification of the Z₂ Z₄-linear Hadamard codes and their automorphism groups

Combinatorics, Coding and Security Group (CCSG)A Z₂ Z₄-linear Hadamard code of length α + 2β = 2t is a binary Hadamard code, which is the Gray map image of a Z₂ Z₄-additive code with α binary coordinates and β quaternary coordinates. It is known that there are exactly ⌊t-1/2⌋ and ⌊t/2⌋ nonequivalent Z₂ Z₄-linear Hadamard codes of length 2t, with α = 0 and α ≠ 0, respectively, for all t ≥ 3. In this paper, it is shown that each Z₂ Z₄-linear Hadamard code with α = 0 is equivalent to a Z₂ Z₄-linear Hadamard code with α ≠ 0, so there are only ⌊t/2⌋ nonequivalent Z₂ Z₄-linear Hadamard codes of length 2t. Moreover, the order of the monomial automorphism group for the Z2Z4-additive Hadamard codes and the permutation automorphism group of the corresponding Z₂ Z₄-linear Hadamard codes are given

On the Kernel of Z2s\mathbb{Z}_{2^s}-Linear Hadamard Codes

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 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 $s\geq 2$, by using also the rank and, in some cases, further computations

On Z8-linear Hadamard codes : rank and classification

The Z2s -additive codes are subgroups of ℤZn2s, and can be seen as a generalization of linear codes over ℤ2 and ℤ4. A Zs-linear Hadamard code is a binary Hadamard code which is the Gray map image of a ℤs -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 computed for s=3. Moreover, it is proved 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

Multiplicative codes of Reed-Muller type

This is a comprehensive study of multiplicative codes of Reed-Muller type and their applications. Our codes apply to the elds of cryptography and coding theory, especially to multiparty computa- tion and secret sharing schemes. We also study the AB method to analyze the minimum distance of linear codes. The multiplicative codes of Reed-Muller type and the AB method are connected when we study the distance and dual distance of a code and its square. Generator matrices for our codes use a combination of blocks, where a block consists of all columns of a given weight. Several interesting linear codes, which are best known linear codes for a given length and dimension, can be constructed in this way.