Celebratio Mathematica — Berlekamp

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E. R. Berlekamp and N. J. A. Sloane: “Restrictions on weight distribution of Reed–Muller codes,” Inform. Control 14 : 5 (May 1969), pp. 442–456. MR 0243897 Zbl 0179.49101 article

N. J. A. Sloane and E. R. Berlekamp: “Weight enumerator for second-order Reed–Muller codes,” IEEE Trans. Information Theory 16 : 6 (November 1970), pp. 745–751. MR 0274196 Zbl 0205.20701 article

E. R. Berlekamp, F. J. MacWilliams, and N. J. A. Sloane: “Gleason’s theorem on self-dual codes,” IEEE Trans. Information Theory 18 : 3 (May 1972), pp. 409–414. MR 0335128 Zbl 0239.94005 article

The weight enumerator of a code is the polynomial \[ W(x,y) = \sum_{r=0}^n A_r x^{n-r} y^r \] where \( n \) denotes the block length and \( A_r \) denotes the number of codewords of weight \( r \). Let \( C \) be a self-dual code over \( \operatorname{GF}(q) \) in which every weight is divisible by \( c \). Then Gleason’s theorem states that

if \( q = 2 \) and \( c = 2 \), the weight enumerator of \( C \) is a sum of products of the polynomials \[ x^2 + y^2 \quad\text{and}\quad x^2 y^2 (x^2 - y^2 )^2 ;\]
if \( q = 2 \) and \( c = 4 \), the weight enumerator is a sum of products of \[ x^8 + 14x^4 y^4 + y^8 \quad\text{and}\quad x^4 y^4 (x^4 - y^4)^4 ;\]
if \( q = 3 \) and \( c = 3 \), the weight enumerator is a sum of products of \[ x^4 + 8xy^3 \quad\text{and}\quad y^3(x^3 - y^3)^3 .\]

In this paper we give several proofs of Gleason’s theorem.

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