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We study binary self-dual codes with a fixed point free automorphism of order three. All binary codes of that type can be obtained by a cubic construction that generalizes Turyn's. We regard such "cubic" codes of length 31 as codes of length l over the ring [F.sub.2] X [F.sub.4]. Classical notions of Type II codes, shadow codes, and weight enumerators are adapted to that ring. Two infinite families of cubic codes are introduced. New external binary codes in lengths [less than or equal to] 66 are constructed by a randomized algorithm. Necessary conditions for the existence of a cubic [72, 36, 16] Type II code are derived. Index Terms--Automorphism group, codes over rings, self-dual codes.

Type II [Z.sub.4]-codes are introduced as self-dual codes over the integers modulo 4 containing the all-one vector and with Euclidean weights multiple of 8. Their weight enumerators are characterized by means of invariant theory. A notion of extremality for the Euclidean weight is introduced. Their binary images under the Gray map are formally self-dual with even weights. Extended quadratic residue [Z.sub.4]-codes are the main example of this family of codes. They are obtained by Hensel lifting of the classical binary quadratic residue codes. Their binary images have good parameters. With every type II [Z.sub.4]-code is associated via construction A modulo 4 an even unimodular lattice (type H lattice). In dimension 32, we construct two unimodular lattices of norm 4 with an automorphism of order 31. One of them is the Barnes-Wall lattice BW32. Index Terms - Codes over rings, lattices, self-dual codes, weight enumerators, [Z.sub.4]-codes.

We construct new self-dual and isodual codes over the integers modulo 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 [equivalent to] -1(mod 8) 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. Index Terms - Codes over rings, self-dual codes, quadratic residue codes, even unimodular lattices, Leech lattice.

We give a simple decoding algorithm to decode linear cyclic codes of odd length over the ring [Mathematical Expression Omitted], where [u.sup.2] = 0. A spectral representation of the cyclic codes over R is given and a BCH-like bound is given for the Lee distance of the codes. The ring R shares many properties of [Z.sub.4] and [F.sub.4] and admits a linear "Gray map." Index Terms - Codes over rings, cyclic codes, Gray map, self-dual codes.