5-Designs from the lifted Golay code over Z4 via an Assmus–Mattson type approach

Recently, Harada showed that the codewords of Hamming weight 10 in the lifted quaternary Golay code form a 5-design. The codewords of Hamming weight 12 in the lifted Golay code are of two symmetric weight enumerator (swe) types. The codewords of each of the two swe types were also shown by Harada to form a 5-design. While Haradaʹs results were obtained via computer search, a subsequent analytical proof of these results appears in a paper by Bonnecaze, Rains and Sole. Here we provide an alternative analytical proof, using an Assmus–Mattson type approach, that the codewords of Hamming weight 12 in the lifted Golay code of each symmetric weight enumerator type, form a 5-design. Also included in the paper is the weight hierarchy of the lifted Golay code. The generalized Hamming weights are used to distinguish between simple 5-designs and those with repeated blocks.