1,922 research outputs found

    On Quaternary linear Reed-Muller codes

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    A la literatura recent hi podem trobar la introducció de noves famílies de codis de Reed- Muller quaternaris lineals RMs. Les imatges d'aquests nous codis a través del mapa de Gray són codis binaris Z4-lineals que comparteixen els paràmetres i les propietats (longitud, dimensió, distància mínima, inclusió, i relació de dualitat) amb la família de codis de Reed- Muller binaris lineals. El kernel d'un codi binari C es defineix com K(C) = {x 2 Zn2 : C + x = C}. La dimensió del kernel és un invariant estructural per els codis binaris equivalents. Part d'aquesta tesi consisteix en establir els valors de la dimensió del kernel per aquestes noves famílies de codis de Reed-Muller Z4-lineals. Tot i que dos codis Z4- lineals no equivalents poden compartir el mateix valor de la dimensió del kernel, en el cas dels codis de Reed-Muller RMs aquest resultat es suficient per donar-ne una classificació completa. Per altra banda, un codi quaternari lineal de Hadamard C, és un codi que un cop li hem aplicat el mapa de Gray obtenim un codi binari de Hadamard. És conegut que els codis de Hadamard quaternaris formen part de les famílies de codis quaternaris de Reed- MullerRMs. Definim el grup de permutacions d'un codi quaternari lineal com PAut(C) = { 2 Sn : (C) = C}. Com a resultat d'aquesta tesi també s'estableix l'ordre dels grups de permutacions de les famílies de codis de Hadamard quaternaris. A més a més, aquests grups són caracteritzats proporcionant la forma dels seus generadors i la forma de les òrbites del grup PAut(C) actuant sobre el codi C. Sabem que el codi dual, en el sentit quaternari, d'un codi de Hadamard és un codi 1-perfecte estès. D'aquesta manera els resultats obtinguts sobre el grup de permutacions es poden transportar a una família de codis quaternaris 1- perfectes estesosRecently, new families of quaternary linear Reed-Muller codes RMs have been introduced. They satisfy that, under the Gray map, the corresponding Z4-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 2 Zn2 : 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 Z4-linear Reed-Muller codes is established. This result is sufficient to give a full classification of these new families of Z4-linear Reed-Muller codes up to equivalence. A quaternary linear Hadamard code C is a code over Z4 that under the Gray map, the corresponding Z4-linear code is a binary Hadamard code. It is well known that quaternary linear Hadamard codes are included in the RMs families of codes. The permutation automorphism group of a quaternary linear code C of length n is defined as PAut(C) = { 2 Sn : (C) = 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

    Peak-to-mean power control and error correction for OFDM transmission using Golay sequences and Reed-Muller codes

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    A coding scheme for OFDM transmission is proposed, exploiting a previously unrecognised connection between pairs of Golay complementary sequences and second-order Reed-Muller codes. The scheme solves the notorious problem of power control in OFDM systems by maintaining a peak-to-mean envelope power ratio of at most 3dB while allowing simple encoding and decoding at high code rates for binary, quaternary or higher-phase signalling together with good error correction

    Peak-to-Mean Power Control in OFDM, Golay Complementary Sequences, and Reed–Muller Codes

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    We present a range of coding schemes for OFDM transmission using binary, quaternary, octary, and higher order modulation that give high code rates for moderate numbers of carriers. These schemes have tightly bounded peak-to-mean envelope power ratio (PMEPR) and simultaneously have good error correction capability. The key theoretical result is a previously unrecognized connection between Golay complementary sequences and second-order Reed–Muller codes over alphabets ℤ2h. We obtain additional flexibility in trading off code rate, PMEPR, and error correction capability by partitioning the second-order Reed–Muller code into cosets such that codewords with large values of PMEPR are isolated. For all the proposed schemes we show that encoding is straightforward and give an efficient decoding algorithm involving multiple fast Hadamard transforms. Since the coding schemes are all based on the same formal generator matrix we can deal adaptively with varying channel constraints and evolving system requirements

    Binary codes : binary codes databases

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    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..

    Q-ary codes

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    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

    A linear construction for certain Kerdock and Preparata codes

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    The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes are shown to be linear over \ZZ_4, the integers  mod  4\bmod~4. The Kerdock and Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over \ZZ_4. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in general are not, nor is the Golay code.Comment: 5 page
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