A table of weight distributions of198minimal cyclic codes is presented.

V. K. Bhargava recently made the following suggestion. LetChe a cyclic code of dimensionp, and block lengthn, wheren = tp, pis a prime. It is sometimes possible to find a code equivalent toCwhich has a generator matrix of the form[I|A_{1}| \cdots |A_{t-1}]where eachA_{i}is ap \times pcirculant matrix. Such a decomposition is useful in encoding and decoding and also in finding the weight distribution ofC. Some general theory as to

A table of primitive binary idempotents of odd block lengthn, 7 \leq n \leq 511, is presented.

Improved bounds forA(n,d), the maximum number of codewords in a (linear or nonlinear) binary code of word lengthnand minimum distanced, and forA(n,d,w), the maximum number of binary vectors of lengthn, distanced, and constant weightwin the rangen \leq 24andd \leq 10are presented. Some of the new values areA

A simple definition of generalized quadratic residue codes, that is, quadratic residue codes of block lengthp^{m}, is given, and an account of many of their properties is presented.

LetF = GF(2), andFG = F[x]/(x^n + 1). FGis the residue class ring of polynomials modx^n + 1. An element ofFGis represented by a polynomial of degree at mostn - 1\begin{equation} c(x) = c_0 + c_1 x + \cdots + c_{n-1} x^{n-1} \end{equation} with coefficients inF. It may also be represented by a polynomial \begin{equation} g(z) = \sum_{j=0}^{n-1} c(\alpha^j)z^j \end{equation} with coefficients inGF(2^m), wheremis the least integer such thatndivides2^m - 1, and\alphais a primitiventh root of unity. Mattson and Solomon [1] introduced this representation in 1961. The new theorem states that \begin{equation} zg'(z) = \frac{g(z)(g(z) + 1)}{z^n +l}. \end{equation} A typical application of this result is as follows. Letn = 2^m - 1, wherem \equiv 1mod 2. Let\mathcal{A}_1be the cyclic code of dimension 2m defined by the property that its check polynomial has zeros\alpha ^{-j}, wherej = 1,2,\cdots,2^{m-1}andj = l,2l,\cdots,2^{m-1} l, l = 2^i + 1. If(i,m) = 1this code has just three nonzero weights, namely,2^{m-1} \pm 2^{(m-1)/2}and2^{m-1}. The weight distribution can then be obtained from the MacWflliams identifies. These conditions are satisfied forn = 31, l = 3,5; n = 127,l= 3,5,9;n = 511, l = 3,5,17; etc. Thus forn= 127, for example, the three codes\mathcal{A}_3,\mathcal{A}_5, \mathcal{A}_9have the same weight distribution, although they are probably not equivalent in the usual sense.

The weight enumerator of a code is the polynomial \begin{equation} W(x,y)= \sum_{r=0}^n A_r x^{n-r} y^r, \end{equation} wherendenotes the block length andA_r, denotes the number of codewords of weightr. LetCbe a self-dual code overGF(q)in which every weight is divisible byc. Then Gleason's theorem states that 1) ifq= 2 andc= 2, the weight enumerator ofCis a sum of products of the polynomialsx^2 + y^2andx^2y^2 (x^2 - y^2 )^2ifq= 2 andc= 4, the weight enumerator is a sum of products ofx^8 + 14x^4 y^4 + y^8andx^4 y^4 (x^4 - y^4)^4; and 3) ifq= 3 andc= 3, the weight enumerator is a sum of products ofx^4 + 8xy^3andy^3(x^3 - y^3)^3. In this paper we give several proofs of Gleason's theorem.

Gleason has recently shown that the weight enumerators of binary and ternary self-dual codes are polynomials in two given polynomials. In this paper it is shown that classical invariant theory permits a straightforward and systematic proof of Gleason's theorems and their generalizations. The joint weight enumerator of two codes (analogous to the joint density function of two random variables) is defined and shown to satisfy a MacWilliams theorem. Invariant theory is then applied to generalize Gleason's theorem to the complete weight enumerator of self-dual codes overGF(3), the Lee metric enumerator overGF(5)(given by Klein in 1884!) and overGF(7)(given by Maschke in 1893!), the Hamming enumerator overGF(q), and overGF(4)with all weights divisible by 2, the joint enumerator of two self-dual codes overGF(2), and a number of other results.