Double Circulant Codes over \mathbb Z 4 \mathbb{Z}_4 and Even Unimodular Lattices
A.R. Calderbank
and N.J.A. Sloane
AT\&T Labs-Research Information Sciences Research Center Murray Hill New Jersey 07974
DOI: 10.1023/A:1008639004036
Abstract
With the help of some new results about weight enumerators of self-dual codes over \mathbb Z 4 \mathbb{Z}_4 we investigate a class of double circulant codes over \mathbb Z 4 \mathbb{Z}_4 , one of which leads to an extremal even unimodular 40-dimensional lattice. It is conjectured that there should be \mathbb Z 4 {\mathbb{Z}}_4 - Leech lattice - invariant theory
Pages: 119–131
Keywords: quarterly code; unimodular lattice; Z4; Leech lattice; invariant theory
Full Text: PDF
References
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8. J.H. Conway and N.J. A. Sloane, “Twenty-three constructions for the Leech lattice,” Proc. Roy. Soc. London, Series A 381 (1982), 275-283. A revised version appears as Chapter 24 of Ref. [9].
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10. J.H. Conway and N.J.A. Sloane, “Self-dual codes over the integers modulo 4,” J. Combinatorial Theory, Series A 62 (1993), 30-45. P1: RSA/SRK P2: RSA/ASH P3: RSA/ASH QC: Journal of Algebraic Combinatorics KL401-02-Calderbank January 30, 1997 20:17 DOUBLE CIRCULANT CODES OVER Z4 AND EVEN UNIMODULAR LATTICES 131
11. G.D. Forney, Jr., N.J.A. Sloane, and M.D. Troff, “The Nordstrom-Robinson code is the binary image of the octacode,” in Coding and Quantization: DIMACS/IEEE Workshop October 19-21, 1992, R. Calderbank, G.D. Forney, Jr., and N. Moayeri (Eds.) Amer. Math. Soc., 1993, pp. 19-26.
12. A.R. Hammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Solé, “The Z4-linearity of Kerdock, Preparata, Goethals and related codes,” IEEE Trans. Inform. Theory 40 (1994), 301-319.
13. M. Klemm, “Selbstduale Codes \ddot uber dem Ring der ganzen Zahlen modulo 4,” Archiv Math. 53 (1989), 201-207.
14. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
15. V. Pless and N.J.A. Sloane, “On the classification and enumeration of self-dual codes,” J. Combinatorial Theory, Series A 18 (1975), 313-335.
16. G.C. Shephard and J.A. Todd, “Finite unitary reflection groups,” Canad. J. Math. 6 (1954), 274-304.
17. N.J.A. Sloane, “Error-correcting codes and invariant theory: New applications of a nineteenth-century technique,” Am. Math. Monthly 84 (1977), 82-107.
18. L. Smith, Polynomial Invariants of Finite Groups, Peters, Wellesley, MA, 1995.
19. B. Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, NY, 1993.
2. A. Bonnecaze, Codes sur des anneaux finis et réseaux arithmétiques, Ph.D. dissertation, University of Nice, Oct. 1995.
3. A. Bonnecaze, A.R. Calderbank, and P. Solé, “Quaternary quadratic residue codes and unimodular lattices,” IEEE Trans. Inform. Theory 41 (1995), 366-377.
4. A. Bonnecaze, P. Solé and B. Mourrain, Quaternary Type II Codes, preprint 1995.
5. W. Bosma and J. Cannon, Handbook of Magma Functions, Math. Dept., Univ. of Sydney, Sydney, Nov. 25, 1994.
6. A.R. Calderbank and N.J.A. Sloane, “Modular and p-adic cyclic codes,” Designs, Codes and Cryptography 6 (1995), 21-35.
7. J.H. Conway, V. Pless, and N.J.A. Sloane, “The binary self-dual codes of length up to 32: A revised enumeration,” J. Combinatorial Theory, Series A 60 (1992), 183-195.
8. J.H. Conway and N.J. A. Sloane, “Twenty-three constructions for the Leech lattice,” Proc. Roy. Soc. London, Series A 381 (1982), 275-283. A revised version appears as Chapter 24 of Ref. [9].
9. J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 2nd edition, 1993.
10. J.H. Conway and N.J.A. Sloane, “Self-dual codes over the integers modulo 4,” J. Combinatorial Theory, Series A 62 (1993), 30-45. P1: RSA/SRK P2: RSA/ASH P3: RSA/ASH QC: Journal of Algebraic Combinatorics KL401-02-Calderbank January 30, 1997 20:17 DOUBLE CIRCULANT CODES OVER Z4 AND EVEN UNIMODULAR LATTICES 131
11. G.D. Forney, Jr., N.J.A. Sloane, and M.D. Troff, “The Nordstrom-Robinson code is the binary image of the octacode,” in Coding and Quantization: DIMACS/IEEE Workshop October 19-21, 1992, R. Calderbank, G.D. Forney, Jr., and N. Moayeri (Eds.) Amer. Math. Soc., 1993, pp. 19-26.
12. A.R. Hammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Solé, “The Z4-linearity of Kerdock, Preparata, Goethals and related codes,” IEEE Trans. Inform. Theory 40 (1994), 301-319.
13. M. Klemm, “Selbstduale Codes \ddot uber dem Ring der ganzen Zahlen modulo 4,” Archiv Math. 53 (1989), 201-207.
14. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
15. V. Pless and N.J.A. Sloane, “On the classification and enumeration of self-dual codes,” J. Combinatorial Theory, Series A 18 (1975), 313-335.
16. G.C. Shephard and J.A. Todd, “Finite unitary reflection groups,” Canad. J. Math. 6 (1954), 274-304.
17. N.J.A. Sloane, “Error-correcting codes and invariant theory: New applications of a nineteenth-century technique,” Am. Math. Monthly 84 (1977), 82-107.
18. L. Smith, Polynomial Invariants of Finite Groups, Peters, Wellesley, MA, 1995.
19. B. Sturmfels, Algorithms in Invariant Theory, Springer-Verlag, NY, 1993.