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 kth roots of unity, we can construct a larger Butson matrix over the ℓth roots of unity for any ℓ dividing k, provided that any prime p dividing k also divides ℓ. We prove that a Zps-additive code with p a prime number is isomorphic as a group to a BH-code over Zps and the image of this BH-code under the Gray map is a BH-code over Zp (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

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 \'{O} Cath\'{a}in and Swartz. That is, we show how, if given a Butson Hadamard matrix over the $k^{\rm th}$ roots of unity, we can construct a larger Butson matrix over the $\ell^{\rm th}$ 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.Comment: 24 pages. Submitted to IEEE Transactions on Information Theor

Quantum Codes from additive constacyclic codes over a mixed alphabet and the MacWilliams identities

Let $\mathbb{Z}_p$ be the ring of integers modulo a prime number $p$ where $p-1$ is a quadratic residue modulo $p$. This paper presents the study of constacyclic codes over chain rings $\mathcal{R}=\frac{\mathbb{Z}_p[u]}{\langle u^2\rangle}$ and $\mathcal{S}=\frac{\mathbb{Z}_p[u]}{\langle u^3\rangle}$. We also study additive constacyclic codes over $\mathcal{R}\mathcal{S}$ and $\mathbb{Z}_p\mathcal{R}\mathcal{S}$ using the generator polynomials over the rings $\mathcal{R}$ and $\mathcal{S},$ respectively. Further, by defining Gray maps on $\mathcal{R}$, $\mathcal{S}$ and $\mathbb{Z}_p\mathcal{R}\mathcal{S},$ we obtain some results on the Gray images of additive codes. Then we give the weight enumeration and MacWilliams identities corresponding to the additive codes over $\mathbb{Z}_p\mathcal{R}\mathcal{S}$. Finally, as an application of the obtained codes, we give quantum codes using the CSS construction.Comment: 22 page

Z2Z4-additive codes

Altres ajuts: UAB PNL2006-13The Combinatoric, Coding and Security Group (CCSG) is a research group in the Department of Information and Communications Engineering (DEIC) at the Universitat Aut'onoma de Barcelona (UAB). The research group CCSG has been uninterruptedly working since 1987 in several projects and research activities on Information Theory, Communications, Coding Theory, Source Coding, Cryptography, Electronic Voting, Network Coding, etc. The members of the group have been producing mainly results on optimal coding. Specifically, the research has been focused on uniformly-packed codes; perfect codes in the Hamming space; perfect codes in distance-regular graphs; the classification of optimal codes of a given length; and codes which are close to optimal codes by some properties, for example, Reed-Muller codes, Preparata codes, Kerdock codes and Hadamard codes. Part of the research developed by CCSG deals with Z2Z4-linear codes. There are no symbolic software to work with these codes, so the members of CCSG have been developing this new package that supports the basic facilities for Z2Z4-additive codes. Specifically, this Magma package generalizes most of the known functions for codes over the ring Z4, which are subgroups of Zn4, to Z2Z4-additive codes, which are subgroups of Zγ2 × Zδ4, maintaining all the functionality for codes over Z4 and adding new functions which, not only generalize the previous ones, but introduce new variants when it is needed. A beta version of this new package for Z2Z4-additive codes and this manual with the description of all functions can be downloaded from the web page http://ccsg.uab.cat. For any comment or further information about this package, you can send an e-mail to [email protected]. The authors would like to thank Lorena Ronquillo, Jaume Pernas, Roger Ten-Valls, and Cristina Diéguez for their contributions developing some parts of this Magma package

Z2Z4-additive codes

Altres ajuts: UAB PNL2006-13The Combinatoric, Coding and Security Group (CCSG) is a research group in the Department of Information and Communications Engineering (DEIC) at the Universitat Aut'onoma de Barcelona (UAB). The research group CCSG has been uninterruptedly working since 1987 in several projects and research activities on Information Theory, Communications, Coding Theory, Source Coding, Cryptography, Electronic Voting, Network Coding, etc. The members of the group have been producing mainly results on optimal coding. Specifically, the research has been focused on uniformly-packed codes; perfect codes in the Hamming space; perfect codes in distance-regular graphs; the classification of optimal codes of a given length; and codes which are close to optimal codes by some properties, for example, Reed-Muller codes, Preparata codes, Kerdock codes and Hadamard codes. Part of the research developed by CCSG deals with Z2Z4-linear codes. There are no symbolic software to work with these codes, so the members of CCSG have been developing this new package that supports the basic facilities for Z2Z4-additive codes. Specifically, this Magma package generalizes most of the known functions for codes over the ring Z4, which are subgroups of Zn4, to Z2Z4-additive codes, which are subgroups of Zγ2 × Zδ4, maintaining all the functionality for codes over Z4 and adding new functions which, not only generalize the previous ones, but introduce new variants when it is needed. A beta version of this new package for Z2Z4-additive codes and this manual with the description of all functions can be downloaded from the web page http://ccsg.uab.cat. For any comment or further information about this package, you can send an e-mail to [email protected]. The authors would like to thank Lorena Ronquillo, Jaume Pernas, Roger Ten-Valls, and Cristina Diéguez for their contributions developing some parts of this Magma package