TY - JOUR

T1 - A New Theorem About the Mattson-Solomon Polynomial and Some Applications

AU - Kerdock, Anthony M.

AU - MacWilliams, F. Jessie

AU - Odlyzko, Andrew M.

PY - 1974/1

Y1 - 1974/1

N2 - Let F = GF(2), and FG = F[x]/(xn + 1). FG is the residue class ring of polynomials mod xn + 1. An element of FG is represented by a polynomial of degree at most n − 1 c(x) = co + c1x + … + cn−1xn−1 with coefficients in F. It may also be represented by a polynomial [formula omitted] with coefficients in GF(2m), where m is the least integer such that n divides 2m − 1, and α is a primitive nth root of unity. Mattson and Solomon [1] introduced this representation in 1961. The new theorem states that [formula omitted] A typical application of this result is as follows. Let n = 2m − 1, where m ≡ 1 mod 2. Let Al be the cyclic code of dimension 2m defined by the property that its check polynomial has zeros α−j, where j = 1,2,⋯, 2m−1 and j = 1,2l,⋯,2m−1 l, l = 2i + 1. If (i,m) = 1 this code has just three nonzero weights, namely, 2m−1 ± 2(m−1)/2 and 2m−1. The weight distribution can then be obtained from the MacWilliams identities. These conditions are satisfied for n = 31, l = 3,5; n = 127, l = 3,5,9; n = 511, l= 3,5,17; etc. Thus for n = 127, for example, the three codes A3, A5, A9 have the same weight distribution, although they are probably not equivalent in the usual sense.

AB - Let F = GF(2), and FG = F[x]/(xn + 1). FG is the residue class ring of polynomials mod xn + 1. An element of FG is represented by a polynomial of degree at most n − 1 c(x) = co + c1x + … + cn−1xn−1 with coefficients in F. It may also be represented by a polynomial [formula omitted] with coefficients in GF(2m), where m is the least integer such that n divides 2m − 1, and α is a primitive nth root of unity. Mattson and Solomon [1] introduced this representation in 1961. The new theorem states that [formula omitted] A typical application of this result is as follows. Let n = 2m − 1, where m ≡ 1 mod 2. Let Al be the cyclic code of dimension 2m defined by the property that its check polynomial has zeros α−j, where j = 1,2,⋯, 2m−1 and j = 1,2l,⋯,2m−1 l, l = 2i + 1. If (i,m) = 1 this code has just three nonzero weights, namely, 2m−1 ± 2(m−1)/2 and 2m−1. The weight distribution can then be obtained from the MacWilliams identities. These conditions are satisfied for n = 31, l = 3,5; n = 127, l = 3,5,9; n = 511, l= 3,5,17; etc. Thus for n = 127, for example, the three codes A3, A5, A9 have the same weight distribution, although they are probably not equivalent in the usual sense.

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U2 - 10.1109/TIT.1974.1055168

DO - 10.1109/TIT.1974.1055168

M3 - Article

AN - SCOPUS:0015959694

SN - 0018-9448

VL - 20

SP - 85

EP - 89

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 1

ER -