Julin-Golay code[1–3] 

One of several nonlinear binary \((12,144,4)\) codes based on the Steiner system \(S(5,6,12)\) [4,5][6; Sec. 2.7][7; Sec. 4] or their shortened versions, the nonlinear \((11,72,4)\), \((10,38,4)\), and \((9,20,4)\) Julin-Golay codes. Several of these codes contain more codewords than linear codes of the same length and distance and yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.

The codewords of the length-12 codes are 132 distinct mod-two pairwise row sums of an \(11\)-dimensional matrix derived from the \(12\)-dimensional Hadamard matrix \(H\) along with their negations, 6 mutually disjoint codewords of weight two, and 6 codewords of weight 10 whose complements are mutually disjoint.

Using Construction A, some Julin-Golay length-12 codes yield 12-dimensional non-lattice sphere packings, collectively called \(P_{12a}\), with kissing number 840 [3][8; pg. 139]. This is the highest known kissing number in that dimension. The length-11 code yields \(P_{11a}\), a non-lattice sphere packing that is the densest known in 11 dimensions. The length-9 code yields a non-lattice sphere packing called \(P_{9a}\) with kissing number 306, the highest known in 9 dimensions.

The Julin-Golay length-12 codes are not to be confused with the Best \((12,144,4)\) code [9], which is not based on a Steiner system [7; Sec. 3].