A linear construction for certain Kerdock and Preparata codes

The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes are shown to be linear over \ZZ_4, the integers $\bmod~4$. The Kerdock and Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over \ZZ_4. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in general are not, nor is the Golay code.Comment: 5 page

The eigenmatrix of the linear association scheme on R(2,m)

AbstractLet R(r,m) be the rth order Reed-Muller code of length 2m. For −1⩽r⩽s⩽m, the action of the general affine group AGL(m,2) on R(s,m)/R(r,m) defines a linear association scheme on R(s,m)/R(r,m). In this paper, we determine the eigenmatrix of the linear association scheme on R(2,m)(=R(2,m)/R(−1,m)). Our approach relies on the Möbius inversion and detailed calculations with the general linear group and the symplectic group over GF(2). As a consequence, we obtain explicit formulas for the weight enumerators of all cosets of R(m−3,m). Such explicit formulas were not available previously

Quantum Goethals-Preparata Codes

We present a family of non-additive quantum codes based on Goethals and Preparata codes with parameters ((2^m,2^{2^m-5m+1},8)). The dimension of these codes is eight times higher than the dimension of the best known additive quantum codes of equal length and minimum distance.Comment: Submitted to the 2008 IEEE International Symposium on Information Theor

Non-Additive Quantum Codes from Goethals and Preparata Codes

We extend the stabilizer formalism to a class of non-additive quantum codes which are constructed from non-linear classical codes. As an example, we present infinite families of non-additive codes which are derived from Goethals and Preparata codes.Comment: submitted to the 2008 IEEE Information Theory Workshop (ITW 2008

Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes

A partition of a finite abelian group gives rise to a dual partition on the character group via the Fourier transform. Properties of the dual partitions are investigated and a convenient test is given for the case that the bidual partition coincides the primal partition. Such partitions permit MacWilliams identities for the partition enumerators of additive codes. It is shown that dualization commutes with taking products and symmetrized products of partitions on cartesian powers of the given group. After translating the results to Frobenius rings, which are identified with their character module, the approach is applied to partitions that arise from poset structures

Non-Additive Quantum Codes from Goethals and Preparata Codes

We extend the stabilizer formalism to a class of non-additive quantum codes which are constructed from non-linear classical codes. As an example, we present infinite families of non-additive codes which are derived from Goethals and Preparata codes.Comment: submitted to the 2008 IEEE Information Theory Workshop (ITW 2008

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page