Quaternary quadratic residue codes and unimodular lattices. (English) Zbl 0822.94009

Bonnecaze, Alexis; Solé, Patrick; Calderbank, A. R.

IEEE Trans. Inf. Theory 41, No. 2, 366-377 (1995).

After the paper by A. R. Hammons et al. [IEEE Trans. Inf. Theory 40, 301-319 (1994; Zbl 0811.94039)]it is clear that considering quaternary codes, i.e. linear codes over \(\mathbb{Z}_ 4\), the ring of integers modulo 4, one can explain many phenomenae arising for nonlinear binary error-correcting codes. In particular, this approach seems to be the most natural one for constructing Nordstrom-Robinson, Kerdock, and Preparata codes and, in certain cases, for establishing their duality. The authors present some fundamental notions concerning quaternary codes (weight enumerators, the MacWilliams identity, the Gray map) and construct several interesting examples. Their main tool is Hensel lifting from \(\mathbb{Z}_ 2\) to \(\mathbb{Z}_ 4\) applied to classical binary quadratic residue codes. Among the resulting quaternary quadratic residue codes one can find some new self-dual and isodual codes.
They also describe some connections with the finite Fourier transform, combinatorial designs, and unimodular lattices. In particular, the quaternary Golay code gives rise to a new construction of the Leech lattice \(\Lambda_{24}\); other ones produce the shorter Leech lattice \(O_{23}\) and the Barnes-Wall lattice \(BW_{32}\).

Reviewer: B.Kunyavskii (Ramat Gan)

MSC:

94B40	Arithmetic codes
11H06	Lattices and convex bodies (number-theoretic aspects)

Keywords:

quaternary codes; weight enumerators; MacWilliams identity; Gray map; Hensel lifting; quaternary quadratic residue codes; quaternary Golay code; Leech lattice; Barnes-Wall lattice; self-dual codes; isodual codes

Citations:

Zbl 0811.94039