A classification of the cosets of the Reed-Muller code

Author:
James A. Maiorana

Journal:
Math. Comp. **57** (1991), 403-414

MSC:
Primary 94B05; Secondary 94-04, 94B35

DOI:
https://doi.org/10.1090/S0025-5718-1991-1079027-8

MathSciNet review:
1079027

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Abstract: The weight distribution of a coset of a Reed-Muller code is invariant under a large transformation group consisting of all affine rearrangements of a vector space with dimension *m*. We discuss a general algorithm that produces an ordered list of orbit representatives for this group action. As a by-product the procedure finds the order of the symmetry group of a coset.

With we can implement the algorithm on a computer and find that there are 150357 equivalence classes. These classes produce 2082 distinct weight distributions. Their symmetry groups have 122 different orders.

**[1]**Elwyn R. Berlekamp and Lloyd R. Welch,*Weight distributions of the cosets of the (32,6) Reed-Muller code*, IEEE Trans. Information Theory**IT-18**(1972), 203–207. MR**0396054****[2]**F. J. MacWilliams and N. J. A. Sloane,*The theory of error-correcting codes*, North-Holland, New York, 1977.**[3]**Michael A. Harrison,*Counting theorems and their applications to classification of switching functions*, Recent Developments in Switching Theory, Academic Press, New York, 1971, pp. 85–120. MR**0280279****[4]***Recent developments in switching theory*, Edited by Amar Mukhopadhyay, Academic Press, New York-London, 1971. MR**0278844****[5]**Charles C. Sims,*Determining the conjugacy classes of a permutation group*, Computers in algebra and number theory (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 191–195. SIAM-AMS Proc., Vol. IV. MR**0338135****[6]**-,*Computation with permutation groups*, Proc. Second Sympos. on Symbolic and Algebraic Manipulation, Assoc. Comput. Mach., New York, 1971.

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DOI:
https://doi.org/10.1090/S0025-5718-1991-1079027-8

Article copyright:
© Copyright 1991
American Mathematical Society