On the Weight Enumerators of the Projections of the 2-adic Golay Code of Length 24 to Z2e

In [2], Calderbank and Sloane investigated codes over the p-adics and examined lifts of codes over Zp to Zpe and to the p-adics. They described the [8, 4, 5] 2-adic Hamming code, the [24, 12, 13] 2-adic Golay code, and the [12, 6, 7] 3-adic Golay code. These three codes are all self-dual and MDS. In [3], Dougherty, Kim, and Park gave the weight distribution of the projections of these three codes to Zpe . For the 2-adic Hamming code and the 3-adic Golay code, they completed the weight distribution. But they did not complete the case of 2-adic Golay code. In this paper, we complete the weight distribution of the projections of the 2-adic Golay code to Zpe for all e ≥ 1. In specific, let Ge be the projections of the 2-adic Golay code to Zpe for some e ≥ 1. Then the weight enumerator of Ge is given by (1) WGe(x, y) = 12 ∑ j=0 ci ( x + (2 − 1)y )j (xy − y2)24−2j . There are 13 unknowns c0, c1, c2, . . . , c12 in (1). Let (A e 0, A e 1, . . . , A e 24) be the weight distribution of Ge. From [2], we know the minimum distance of Ge is eight for all e ≥ 1, and from [3] we know A8 = 759 and A9 = 0 for all e ≥ 1. So that if we know A10, A e 11, and A e 12, then we can determine WGe(x, y). From [3], we also know that A10, A e 11, A e 12 remain constant for e ≥ 7. In summary, we only have to calculate A10, A e 11, and A e 12 for e = 3, 4, 5, 6, 7 to complete the weight distribution of Ge for all e ≥ 1. The object of this paper is to compute these values. All the computation of this paper was made using Magma [1] with a 2.3GHz and 3.0GB RAM PC. We will also discuss the algorithmic aspect of the program written in MAGMA. 2010 Mathematics Subject Classification. 94B05.