Alternative names: Mod-four lattice.
Description
A lattice that is constructed from a linear code over \(\mathbb{Z}_4\) using Construction \(A_4\).
Construction \(A_4\): Construction \(A_4\) converts a linear code over \(\mathbb{Z}_4\) into a lattice [1; Sec. 12.5.3]. Each codeword \(c\) of the code is mapped to an infinite set of points \(x\) such that \(2x = c\) modulo four.
Cousins
- Linear code over \(\mathbb{Z}_4\)— Every linear code over \(\mathbb{Z}_4\) yields a lattice under Construction \(A_4\) [1; Sec. 12.5.3].
- Unimodular lattice— Type I (type II) codes over \(\mathbb{Z}_4\) yield type I (type II) lattices under Construction \(A_4\) [1; Thm. 12.5.12].
Member of code lists
Primary Hierarchy
Parents
Construction \(A_4\) code
Children
The union of RM\((1,5)\) and 2RM\((3,5)\) codes yields a Type II self-dual linear code over \(\mathbb{Z}_4\) that then gives rise to the \(B_{32}\) Barnes-Wall lattice via Construction \(A_4\) [2,3].
The Leech lattice can be constructed from pseudo Golay codes via Construction \(A_4\) [4,5]. The Leech lattice can be obtained by lifting the Golay code to \(\mathbb{Z}_4\) [6], appending a parity check, and applying Construction \(A_4\) [7] (see also [8,9]). Half of the lattice can be obtained in a different construction [10; Exam. 10.7.3].
Niemeier lattices can be constructed from quaternary codes over \(\mathbb{Z}_4\) via Construction \(A_4\) [11]. These codes are the Harada-Kitazume codes [12].
The octacode yields the \(E_8\) Gosset lattice via Construction \(A_4\) [3,7][1; Exam. 12.5.13].
References
- [1]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [2]
- P. Sole, "Generalized theta functions for lattice vector quantization", in Coding and Quantization, DIMACS Series in Dr,crete Mathenulies and Theoretical Computer Science, vol. 14. Providence, RH: American Math. Soc., 1993, pp. 27-32.
- [3]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
- [4]
- E. Rains, “Optimal self-dual codes over Z4”, Discrete Mathematics 203, 215 (1999) DOI
- [5]
- G. W. Moore and R. K. Singh, “Beauty And The Beast Part 2: Apprehending The Missing Supercurrent”, (2023) arXiv:2309.02382
- [6]
- A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
- [7]
- A. Bonnecaze and P. Solé, “Quaternary constructions of formally self-dual binary codes and unimodular lattices”, Lecture Notes in Computer Science 194 (1994) DOI
- [8]
- “Twenty-three constructions for the Leech lattice”, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 381, 275 (1982) DOI
- [9]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [10]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [11]
- A. Bonnecaze, P. Gaborit, M. Harada, M. Kitazume, and P. Solé, “Niemeier lattices and Type II codes over Z4”, Discrete Mathematics 205, 1 (1999) DOI
- [12]
- M. Harada and M. Kitazume, “Z4-Code Constructions for the Niemeier Lattices and their Embeddings in the Leech Lattice”, European Journal of Combinatorics 21, 473 (2000) DOI
Page edit log
- Victor V. Albert (2025-04-03) — most recent
Cite as:
“Construction \(A_4\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/construction_a4