Binary codes : binary codes databases

Combinatorics, Coding and Security Group (CCSG).The research group CCSG (Combinatorics, Coding and Security Group) is one of the research groups in the dEIC (Department of Information and Communications Engineering) at the UAB (Universitat Autònoma de Barcelona) in Spain. From 1987 the team CCSG has been uninterruptedly working in several projects and research activities on Information Theory, Communications, Coding Theory, Source Coding, Teledetection, Cryptography, Electronic Voting, e-Auctions, Mobile Agents, etc. The more important know-how of CCSG is about algorithms for forward error correction (FEC), such as Golay codes, Hamming product codes, Reed-Solomon codes, Preparata and Preparata-like codes, (extended) nonlinear 1-perfect codes, Z4-linear codes, Z2Z4-linear codes, etc.; computations of the rank and the dimension of the kernel for nonlinear codes as binary 1-perfect codes, q-ary 1-perfect codes, Preparata codes, Hadamard codes, Kerdock codes, quaternary Reed-Muller codes, etc.; the existence and structural properties for 1-perfect codes, uniformly packed codes, completely regular codes, completely transitive codes, etc..

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

Codes over Z4 and permutation decoding of linear codes

Aquest treball es va publicar com "Linear Codes over the Integer Residue Ring Z4" in Handbook of magma funcions / edited by John Cannon, Wieb Bosma, Claus Fieker and Allan Steel (2017), p. 5575-5616

Q-ary codes

The research group CCSG (Combinatorics, Coding and Security Group) is one of the research groups in the dEIC (Department of Information and Communications Engineering) at the UAB (Universitat Aut'onoma de Barcelona) in Spain. From 1987 the team CCSG has been uninterruptedly working in several projects and research activities on Information Theory, Communications, Coding Theory, Source Coding, Teledetection, Cryptography, Electronic Voting, e-Auctions, Mobile Agents, etc. The more important know-how of CCSG is about algorithms for forward error correction (FEC), such as Golay codes, Hamming product codes, Reed-Solomon codes, Preparata and Preparata-like codes, (extended) nonlinear 1-perfect codes, Z4-linear codes, Z2Z4-linear codes, etc.; computations of the rank and the dimension of the kernel for nonlinear codes as binary 1-perfect codes, q-ary 1-perfect codes, Preparata codes, Hadamard codes, Kerdock codes, quaternary Reed-Muller codes, etc.; the existence and structural properties for 1-perfect codes, uniformly packed codes, completely regular codes, completely transitive codes, etc

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