Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras

We consider a class of map, recently derived in the context of cluster mutation. In this paper we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson commuting functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility of a pair of these functions gives rise to Liouville's equation and the map plays the role of a B\"acklund transformation.Comment: 17 pages, 7 figures. Corrected typos and updated reference detail

The invariants of the Clifford groups

The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not 3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an extraspecial group of order 2^(1+2m) extended by an orthogonal group). This group and its complex analogue CC_m have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge's 1996 result that the space of invariants for C_m of degree 2k is spanned by the complete weight enumerators of the codes obtained by tensoring binary self-dual codes of length 2k with the field GF(2^m); these are a basis if m >= k-1. We also give new constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix [2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power of M, and C_m is the automorphism group of this tensor power. Also, if C is a binary self-dual code not generated by vectors of weight 2, then C_m is precisely the automorphism group of the complete weight enumerator of the tensor product of C and GF(2^m). There are analogues of all these results for the complex group CC_m, with ``doubly-even self-dual code'' instead of ``self-dual code''.Comment: Latex, 24 pages. Many small improvement

On Asymmetric Coverings and Covering Numbers

An asymmetric covering D(n,R) is a collection of special subsets S of an n-set such that every subset T of the n-set is contained in at least one special S with |S| - |T| <= R. In this paper we compute the smallest size of any D(n,1) for n <= 8. We also investigate ``continuous'' and ``banded'' versions of the problem. The latter involves the classical covering numbers C(n,k,k-1), and we determine the following new values: C(10,5,4) = 51, C(11,7,6,) =84, C(12,8,7) = 126, C(13,9,8)= 185 and C(14,10,9) = 259. We also find the number of nonisomorphic minimal covering designs in several cases.Comment: 11 page

The Lattice of N-Run Orthogonal Arrays

If the number of runs in a (mixed-level) orthogonal array of strength 2 is specified, what numbers of levels and factors are possible? The collection of possible sets of parameters for orthogonal arrays with N runs has a natural lattice structure, induced by the ``expansive replacement'' construction method. In particular the dual atoms in this lattice are the most important parameter sets, since any other parameter set for an N-run orthogonal array can be constructed from them. To get a sense for the number of dual atoms, and to begin to understand the lattice as a function of N, we investigate the height and the size of the lattice. It is shown that the height is at most [c(N-1)], where c= 1.4039... and that there is an infinite sequence of values of N for which this bound is attained. On the other hand, the number of nodes in the lattice is bounded above by a superpolynomial function of N (and superpolynomial growth does occur for certain sequences of values of N). Using a new construction based on ``mixed spreads'', all parameter sets with 64 runs are determined. Four of these 64-run orthogonal arrays appear to be new.Comment: 28 pages, 4 figure

Quantum Error Correction and Orthogonal Geometry

A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1 error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors, and 1 to 29 qubits correcting 5 errors.Comment: RevTex, 4 pages, no figures, submitted to Phys. Rev. Letters. We have changed the statement of Theorem 2 to correct it -- we now get worse rates than we previously claimed for our quantum codes. Minor changes have been made to the rest of the pape