The ternary Golay code, the integers mod 9, and the Coxeter-Todd lattice

The 12-dimensional Coxeter-Todd lattice can be obtained by lifting the ternary Golay code to a code over the integers mod 9 and applying Construction A. Several recent papers have pointed out connections between codes over & and lattices [l], [3]. It is the a m of this correspondence to show that the Coxeter-Todd lattice K12 ([4], [5, ch. 4, sec. 91) can be obtained in a similar way from a code over Eg. Let +, = z fmz denote the ring of integers mod m. Let C be a cyclic code of length n over &, where p is a pnme such that ( n , p ) = 1, with generator polynomial g(z). If a is a primitive element of &, let S denote the set of s such that 0’ is a root of g(z). ‘it is pointed out in [2] that C is self-orthogonal if and only if S U (-S) contains representatives from every residue class modn. Furthermore, for a = 2,3, . . . , exactly the same condition is also necessary and sufficient for the Hensel lift (cf. [6]) of C to a cyclic code over Zp to be self-orthogonal mod p a . Self-orthogonality of the lifted code is a purely combinatorial property of the roots of the original code. Consider now the [ll, 5,6] self-orthogonal ternary Golay code, with generator polynomial g(z) = z6 z5 E4 zs + x2 + 1 ([7, p. 4821). This lifts to G ( ~ ) =2 + 2z5 4x4 + zZ3 + xz 3x + 1 a divisor of zil 1 mod 9, which by the previous remark generates a self-orthogonal code over &. Appending a 0 to these words and adjoining the all-1’s vector as an additional generator, we obtain a self-dual code 4 of length 12 over &. By applying Construction A (cf. [5, chs. 5, 7]) , that is, taking all vectors in El2 that are congruent to any codeword of 6 mod 9, and rescaling by dividing by &, we obtain the Coxeter-Todd lattice: a 12-dimensional lattice of determinant 729, minimal squared length 4, and 756 minimal vectors. We omit the straightforward proof that this is isomorphic to the standard construction from a code (the “hexacode”) of length 6 over GF(4) ([4], [ 5 , p. 1281, [SI).