The Weight Distributions of Cosets of the Second-Order Reed-Muller Code of Length 128 in the Third-Order Reed-Muller Code of Length 128

Tadao KASAMI; Toru FUJIWARA; Yoshihisa DESAKI

The Weight Distributions of Cosets of the Second-Order Reed-Muller Code of Length 128 in the Third-Order Reed-Muller Code of Length 128

Tadao KASAMI, Toru FUJIWARA, Yoshihisa DESAKI

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In this paper cosets of the second order Reed-Muller code of length 2^m, denoted RM_m,2, in the third order Reed-Muller code of the same length, denoted RM_m,3, are studied. The set of cosets, RM_m,3/RM_m,2 is partitioned into blocks. Two cosets are in the same block, if and only if there is a transformation in the general linear group by which one coset is transformed into the other. Two cosets in the same block have the same weight distribution. For the code length less than or equal to 128, the representative coset leader of each block is presented and the weight distribution of cosets in the block is computed. By using these results, the extended code of a cyclic code of length 128 between RM_7,2 and RM_7,3 can be decomposed into a set of cosets in RM_7,3/RM_7,2, and its weight distribution can be derived. Several cyclic codes to length 127 are shown to be equivalent and some new linear unequal error protection codes are found.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E79-A No.4 pp.600-608

Publication Date: 1996/04/25

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Type of Manuscript: PAPER

Category: Information Theory and Coding Theory

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Tadao KASAMI, Toru FUJIWARA, Yoshihisa DESAKI, "The Weight Distributions of Cosets of the Second-Order Reed-Muller Code of Length 128 in the Third-Order Reed-Muller Code of Length 128" in IEICE TRANSACTIONS on Fundamentals, vol. E79-A, no. 4, pp. 600-608, April 1996, doi: .
Abstract: In this paper cosets of the second order Reed-Muller code of length 2^m, denoted RM_m,2, in the third order Reed-Muller code of the same length, denoted RM_m,3, are studied. The set of cosets, RM_m,3/RM_m,2 is partitioned into blocks. Two cosets are in the same block, if and only if there is a transformation in the general linear group by which one coset is transformed into the other. Two cosets in the same block have the same weight distribution. For the code length less than or equal to 128, the representative coset leader of each block is presented and the weight distribution of cosets in the block is computed. By using these results, the extended code of a cyclic code of length 128 between RM_7,2 and RM_7,3 can be decomposed into a set of cosets in RM_7,3/RM_7,2, and its weight distribution can be derived. Several cyclic codes to length 127 are shown to be equivalent and some new linear unequal error protection codes are found.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e79-a_4_600/_p

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@ARTICLE{e79-a_4_600,
author={Tadao KASAMI, Toru FUJIWARA, Yoshihisa DESAKI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={The Weight Distributions of Cosets of the Second-Order Reed-Muller Code of Length 128 in the Third-Order Reed-Muller Code of Length 128},
year={1996},
volume={E79-A},
number={4},
pages={600-608},
abstract={In this paper cosets of the second order Reed-Muller code of length 2^m, denoted RM_m,2, in the third order Reed-Muller code of the same length, denoted RM_m,3, are studied. The set of cosets, RM_m,3/RM_m,2 is partitioned into blocks. Two cosets are in the same block, if and only if there is a transformation in the general linear group by which one coset is transformed into the other. Two cosets in the same block have the same weight distribution. For the code length less than or equal to 128, the representative coset leader of each block is presented and the weight distribution of cosets in the block is computed. By using these results, the extended code of a cyclic code of length 128 between RM_7,2 and RM_7,3 can be decomposed into a set of cosets in RM_7,3/RM_7,2, and its weight distribution can be derived. Several cyclic codes to length 127 are shown to be equivalent and some new linear unequal error protection codes are found.},
keywords={},
doi={},
ISSN={},
month={April},}

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TY - JOUR
TI - The Weight Distributions of Cosets of the Second-Order Reed-Muller Code of Length 128 in the Third-Order Reed-Muller Code of Length 128
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 600
EP - 608
AU - Tadao KASAMI
AU - Toru FUJIWARA
AU - Yoshihisa DESAKI
PY - 1996
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E79-A
IS - 4
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - April 1996
AB - In this paper cosets of the second order Reed-Muller code of length 2^m, denoted RM_m,2, in the third order Reed-Muller code of the same length, denoted RM_m,3, are studied. The set of cosets, RM_m,3/RM_m,2 is partitioned into blocks. Two cosets are in the same block, if and only if there is a transformation in the general linear group by which one coset is transformed into the other. Two cosets in the same block have the same weight distribution. For the code length less than or equal to 128, the representative coset leader of each block is presented and the weight distribution of cosets in the block is computed. By using these results, the extended code of a cyclic code of length 128 between RM_7,2 and RM_7,3 can be decomposed into a set of cosets in RM_7,3/RM_7,2, and its weight distribution can be derived. Several cyclic codes to length 127 are shown to be equivalent and some new linear unequal error protection codes are found.
ER -