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

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

A Specialized Court for Social Security? A Critique of Recent Proposals

In this Article Professor Rains evaluates the recent proposals for the creation of a Social Security Court. He evaluates the existing administrative and judicial system for the review of social security claims in light of recent problems. Finally, Professor Rains suggests that many of the present difficulties with the system can be solved by reform of the Social Security Administration\u27s review process rather than creation of an Article I court

Increasing Subsequences and the Classical Groups

We show that the moments of the trace of a random unitary matrix have combinatorial interpretations in terms of longest increasing subsequences of permutations. To be precise, we show that the 2n-th moment of the trace of a random k-dimensional unitary matrix is equal to the number of permutations of length n with no increasing subsequence of length greater than k. We then generalize this to other expectations over the unitary group, as well as expectations over the orthogonal and symplectic groups. In each case, the expectations count objects with restricted "increasing subsequence" length