Universal codes and unimodular lattices

Robin Chapman; Patrick Solé

Journal de théorie des nombres de Bordeaux (1996)

  • Volume: 8, Issue: 2, page 369-376
  • ISSN: 1246-7405

Abstract

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Binary quadratic residue codes of length p + 1 produce via construction B and density doubling type II lattices like the Leech. Recently, quaternary quadratic residue codes have been shown to produce the same lattices by construction A modulo 4 . We prove in a direct way the equivalence of these two constructions for p ≤ 31 . In dimension 32, we obtain an extremal lattice of type II not isometric to the Barnes-Wall lattice B W 32 . The equivalence between construction B modulo 4 plus density doubling and construction A modulo 8 is also considered. In dimension 48 they both led to a new description of the extremal type II lattice P 48 q .

How to cite

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Chapman, Robin, and Solé, Patrick. "Universal codes and unimodular lattices." Journal de théorie des nombres de Bordeaux 8.2 (1996): 369-376. <http://eudml.org/doc/247828>.

@article{Chapman1996,
abstract = {Binary quadratic residue codes of length $p + 1$ produce via construction $B$ and density doubling type II lattices like the Leech. Recently, quaternary quadratic residue codes have been shown to produce the same lattices by construction $A$ modulo $4$. We prove in a direct way the equivalence of these two constructions for $p \le 31$. In dimension 32, we obtain an extremal lattice of type II not isometric to the Barnes-Wall lattice $BW_\{32\}$. The equivalence between construction $B$ modulo $4$ plus density doubling and construction $A$ modulo $8$ is also considered. In dimension 48 they both led to a new description of the extremal type II lattice $P_\{48q\}$.},
author = {Chapman, Robin, Solé, Patrick},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {quadratic residue codes; lattices; construction A; construction B; density doubling; unimodular lattices},
language = {eng},
number = {2},
pages = {369-376},
publisher = {Université Bordeaux I},
title = {Universal codes and unimodular lattices},
url = {http://eudml.org/doc/247828},
volume = {8},
year = {1996},
}

TY - JOUR
AU - Chapman, Robin
AU - Solé, Patrick
TI - Universal codes and unimodular lattices
JO - Journal de théorie des nombres de Bordeaux
PY - 1996
PB - Université Bordeaux I
VL - 8
IS - 2
SP - 369
EP - 376
AB - Binary quadratic residue codes of length $p + 1$ produce via construction $B$ and density doubling type II lattices like the Leech. Recently, quaternary quadratic residue codes have been shown to produce the same lattices by construction $A$ modulo $4$. We prove in a direct way the equivalence of these two constructions for $p \le 31$. In dimension 32, we obtain an extremal lattice of type II not isometric to the Barnes-Wall lattice $BW_{32}$. The equivalence between construction $B$ modulo $4$ plus density doubling and construction $A$ modulo $8$ is also considered. In dimension 48 they both led to a new description of the extremal type II lattice $P_{48q}$.
LA - eng
KW - quadratic residue codes; lattices; construction A; construction B; density doubling; unimodular lattices
UR - http://eudml.org/doc/247828
ER -

References

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  1. [1] A. Bonnecaze, P. Solé, C. Bachoc, B. Mourrain, 'Type II Quaternary Codes' IEEE Trans. Inform. Theory, submitted (1995). Zbl0898.94009
  2. [2] A. Bonnecaze, P. Solé & A.R. Calderbank, 'Quaternary quadratic residue codes and unimodular lattices', IEEE Trans. Inform. Theory, vol. 41, pp. 366-377, March 1995. Zbl0822.94009MR1326285
  3. [3] A.R. Calderbank, private communication (1995). 
  4. [4] A.R. Calderbank, G. MacGuire, P.V. Kumar, T. Helleseth, Cyclic Codes over Z4, Locator polynomials, and Newton identities, preprint (1995). MR1326293
  5. [5] A.R. Calderbank, N.J.A. Sloane, 'Double Circulant Codes over Z4 and Unimodular Lattices', J. of Algebraic Combinatorics, submitted. Zbl0881.94026
  6. [6] J.H. Conway & N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1988. Zbl0634.52002MR920369
  7. [7] G.D. Forney, 'Coset Codes II: Binary Lattices and related codes', IEEE Trans. Information Th. IT-34 (1988) 1152-1187. Zbl0665.94019MR987662
  8. [8] H. Koch & B.B. Venkov, Ueber Ganzhalige Unimodulare Euklidische Gitter, Crelle398 (1989) 144-168. Zbl0667.10020MR998477
  9. [9] P. Loyer, P. Solé, 'Les Réseaux BW32 et U32 sont équivalents', J. de Th. des Nombres de Bordeaux6 (1994) 359-362. Zbl0818.11027MR1360650
  10. [10] F.J. MacWilliams, N.J.A. Sloane, The theory of error correcting codesNorth-Holland (1977). Zbl0657.94010
  11. [11] V. Pless, Z. Qian, 'Cyclic Codes and Quadratic Residue Codes over Z4 ', IEEE Trans. Information Theory submitted. Zbl0859.94018MR1426232
  12. [12] H-G. Quebbemann, 'Zur Klassifikation unimodularer Gitter mit Isometrie von Primzahlordnung', Crelle326 (1981) 158-170. Zbl0452.10027MR622351
  13. [13] R. Schulze-Pillot, 'Quadratic Residue Codes and Cyclotomic lattices ', Arch. Math., Vol. 60 (1993) 40-65. Zbl0792.11008MR1193092

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