Christine Bachoc

Type II codes over Z/sub 4/

The conditions satisfied by the weight enumerator of self-dual codes, defined over the ring of integers module four, have been studied by Klemm (1989), then by Conway and Sloane (1993). The MacWilliams (1977) transform determines a group of substitutions, each of which fixes the weight enumerator of a self-dual code. This weight enumerator belongs to the ring of polynomials fixed by the group of substitutions, called the ring R of invariants. Among all of the quaternary self-dual codes, some have the property that all euclidean weights are multiples of 8. These codes are called type II codes by analogy with the binary case. An upper bound on their minimum euclidean weight is given, thereby leading to a natural notion of extremality akin to similar concepts for type II binary codes and type II lattices. The most interesting examples of type II codes are perhaps the extended quaternary quadratic residue codes. This class of codes includes the octacode [8, 4, 6] and the lifted Golay [24, 12, 12]. Other classes of interest comprise a multilevel construction from binary Reed-Muller and lifted double circulant codes.

Applications of Coding Theory to the Construction of Modular Lattices

We study self-dual codes over certain finite rings which are quotients of quadratic imaginary fields or of totally definite quaternion fields over Q. A natural weight taking two different nonzero values is defined over these rings; using invariant theory, we give a basis for the space of invariants to which belongs the three variables weight enumerator of a self-dual code. A general bound for the weight of such codes is derived. We construct a number of extremal self-dual codes, which are the codes reaching this bound, and derive some extremal lattices of levell=2, 3, 7 and minimum 4, 6, 8.

On Harmonic Weight Enumerators of Binary Codes

We define some new polynomials associated to a linear binary code and a harmonic function of degree k. The case k=0 is the usual weight enumerator of the code. When divided by (xy)k, they satisfy a MacWilliams type equality. When applied to certain harmonic functions constructed from Hahn polynomials, they can compute some information on the intersection numbers of the code. As an application, we classify the extremal even formally self-dual codes of length 12.

Linear programming bounds for codes in grassmannian spaces

In this paper, we develop the linear programming method to obtain bounds for the cardinality of Grassmannian codes endowed with the chordal distance. We obtain a bound and its asymptotic version that generalize the well-known bound for codes in the real projective space obtained by Kabatyanskiy and Levenshtein, and improve the Hamming bound for sufficiently large minimal distances

Invariant Semidefinite Programs

This chapter provides the reader with the necessary background for dealing with semidefinite programs which have symmetry. The basic theory is given and it is illustrated in applications from coding theory, combinatorics, geometry, and polynomial optimization.

Designs in Grassmannian Spaces and Lattices

AbstractWe introduce the notion of a t-design on the Grassmann manifold

Tight p-fusion frames

\mathcal{G}_{m,n}

Bounds for projective codes from semidefinite programming

of the m-subspaces of the Euclidean space