Quaternary quadratic residue codes and unimodular lattices

We construct new self-dual and isodual codes over the integers module 4. The binary images of these codes under the Gray map are nonlinear, but formally self-dual. The construction involves Hensel lifting of binary cyclic codes. Quaternary quadratic residue codes are obtained by Hensel lifting of the classical binary quadratic residue codes. Repeated Hensel lifting produces a universal code defined over the 2-adic integers. We investigate the connections between this universal code and the codes defined over Z/sub 4/, the composition of the automorphism group, and the structure of idempotents over Z/sub 4/. We also derive a square root bound on the minimum Lee weight, and explore the connections with the finite Fourier transform. Certain self-dual codes over Z/sub 4/ are shown to determine even unimodular lattices, including the extended quadratic residue code of length q+1, where q/spl equiv/-1(mod8) is a prime power. When q=23, the quaternary Golay code determines the Leech lattice in this way. This is perhaps the simplest construction for this remarkable lattice that is known. >

[1]  Morris Plotkin,et al.  Binary codes with specified minimum distance , 1960, IRE Trans. Inf. Theory.

[2]  H. Mattson,et al.  New 5-designs , 1969 .

[3]  N. J. A. Sloane,et al.  New family of single-error correcting codes , 1970, IEEE Trans. Inf. Theory.

[4]  Vera Pless,et al.  A classification of self-orthogonal codes over GF(2) , 1972, Discret. Math..

[5]  H. Ward.  Quadratic residue codes and symplectic groups , 1974 .

[6]  E. Shult,et al.  Equiangular lines, the Graph Extension Theorem, and transfer in triply transitive groups , 1974 .

[7]  P. Camion.  Global Quadratic Abelian Codes , 1975 .

[8]  D. E. Taylor.  Regular 2‐Graphs , 1977 .

[9]  Jacobus H. van Lint,et al.  Generalized quadratic residue codes , 1978, IEEE Trans. Inf. Theory.

[10]  G. Beenker.  On double circulant codes , 1980 .

[11]  R. Calderbank,et al.  Multiplying Vectors in Binary Quadratic Residue Codes , 1982 .

[12]  Jacobus H. van Lint,et al.  Introduction to Coding Theory , 1982 .

[13]  A. Calderbank,et al.  A global code invariant under the Higman—Sims group , 1982 .

[14]  A. Robert Calderbank.  A square root bound on the minimum weight in quasi-cyclic codes , 1983, IEEE Trans. Inf. Theory.

[15]  Vera Pless,et al.  Duadic Codes , 1984, IEEE Trans. Inf. Theory.

[16]  Hanfried Lenz,et al.  Design theory , 1985 .

[17]  N. J. A. Sloane,et al.  Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice , 1986, IEEE Trans. Inf. Theory.

[18]  G. David Forney,et al.  Coset codes-II: Binary lattices and related codes , 1988, IEEE Trans. Inf. Theory.

[19]  Gordon R. Lang,et al.  A Leech lattice modem , 1989, IEEE J. Sel. Areas Commun..

[20]  Michael Klemm.  Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4 , 1989 .

[21]  W. Heise,et al.  Topics in Algebraic Coding Theory , 1990 .

[22]  Richard E. Blahut.  The Gleason-Prange theorem , 1991, IEEE Trans. Inf. Theory.

[23]  Patrick Solé.  Generalized Theta Functions for Lattice Vector Quantization , 1992, Coding And Quantization.

[24]  N. J. A. Sloane,et al.  Self-Dual Codes over the Integers Modulo 4 , 1993, J. Comb. Theory, Ser. A.

[25]  Patrick Solé,et al.  Quaternary constructions of formally self-dual binary codes and unimodular lattices , 1993, Algebraic Coding.

[26]  Andries E. Brouwer,et al.  A sharpening of the Johnson bound for binary linear codes and the nonexistence of linear codes with preparata parameters , 1993, Des. Codes Cryptogr..

[27]  P. Solé.  Generalised Theta Functionsb for Lattice Vector Quantization , 1993, Proceedings. IEEE International Symposium on Information Theory.

[28]  Ofer Amrani,et al.  The Leech lattice and the Golay code: bounded-distance decoding and multilevel constructions , 1994, IEEE Trans. Inf. Theory.

[29]  N. J. A. Sloane,et al.  The Z4-linearity of Kerdock, Preparata, Goethals, and related codes , 1994, IEEE Trans. Inf. Theory.

[30]  N. J. A. Sloane,et al.  Modular andp-adic cyclic codes , 1995, Des. Codes Cryptogr..