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Ministerio de Educación, Cultura y Deporte - Gobierno de España

Título: ON QUATERNARY LINEAR REED-MULLER CODES
Autor: Pernas Vallès, Jaume
Universidad: Universidad Autónoma de Barcelona
Departamento: INGENIERIA DE LA INFORMACION Y DE LAS COMUNICACIONES
Fecha de Lectura: 06/06/2012
Mención Europeo / Mención Internacional: Concedido
Programa de doctorado: Doctorado en Informàtica
Dirección:
- VILLANUEVA GAY, MERCE (Director)
- PUJOL CAPDEVILA, JAUME (Codirector)
Tribunal:
- Borges Ayats, Joaquim (presidente)
- Dougherty, Steven (secretario)
- Storme, Leo (vocal)
Descriptores:
- INFORMATICA
Localización: UAB

2012pernaonqua.pdf

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Resumen:
Recently, new families of quaternary linear Reed-Muller codes $\RM_s$ have been introduced. They satisfy that, under the Gray map, the corresponding $\Z_4$-linear codes have the same parameters and properties (length, dimension, minimum distance, inclusion, and duality relation) as the codes of the binary linear Reed-Muller family. The kernel of a binary code $C$ is $K(C)=\{x\in\Z_2^n \;:\; C+x=C\}$. The dimension of the kernel is a structural invariant for equivalent binary codes. In this work, the dimension of the kernel for these new families of $\Z_4$-linear Reed-Muller codes is established. This result is sufficient to give a full classification of these new families of $\Z_4$-linear Reed-Muller codes up to equivalence. A quaternary linear Hadamard code $\mathcal{C}$ is a code over $\mathbb{Z}_4$ that under the Gray map, the corresponding $\Z_4$-linear code is a binary Hadamard code. It is well known that quaternary linear Hadamard codes are included in the $\RM_s$ families of codes. The permutation automorphism group of a quaternary linear code $\mathcal{C}$ of length $n$ is defined as $\mbox{PAut}(\mathcal{C})=\{\sigma \in S_n : \sigma(\mathcal{C}) = \mathcal{C} \}$. In this dissertation, the order of the permutation automorphism group of all quaternary linear Hadamard codes is established. Moreover, these groups are completely characterized by providing their generators and also by computing the orbits of their action on $\C$. Since the dual of a Hadamard code is an extended $1$-perfect code in the quaternary sense, the permutation automorphism group of the quaternary linear extended $1$-perfect codes is also established.

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