## Description

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].

## Parent

## Cousins

- Sphere packing — Using Construction \(A\), the Julin-Golay codes yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.
- Construction-\(A\) code — Using Construction \(A\), the Julin-Golay codes yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.
- \(q\)-ary code over \(\mathbb{Z}_q\) — Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map [7].
- Gray code — Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map [7].

## References

- [1]
- M. Golay, “Binary coding”, Transactions of the IRE Professional Group on Information Theory 4, 23 (1954) DOI
- [2]
- D. Julin, “Two improved block codes (Corresp.)”, IEEE Transactions on Information Theory 11, 459 (1965) DOI
- [3]
- J. Leech and N. J. A. Sloane, “Sphere Packings and Error-Correcting Codes”, Canadian Journal of Mathematics 23, 718 (1971) DOI
- [4]
- J. A. Barrau, On the combinatory problem of Steiner, Proc. Section of Sciences, Koninklijke Akademie van Wetenschappen te Amsterdam 11 (1908), 352–360.
- [5]
- J. Leech, “Some Sphere Packings in Higher Space”, Canadian Journal of Mathematics 16, 657 (1964) DOI
- [6]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [7]
- J. H. Conway and N. J. A. Sloane, “Quaternary constructions for the binary single-error-correcting codes of Julin, Best and others”, Designs, Codes and Cryptography 4, 31 (1994) DOI
- [8]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [9]
- Best, M.R. 1978. Binary codes with minimum distance four. Report ZW 112/78, Math Centrum, Amsterdam.

## Page edit log

- Victor V. Albert (2023-03-31) — most recent

## Cite as:

“Julin-Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/julin12