Balanced generalized weighing matrices are applied for constructing a family of symmetric designs with parameters (1 + qr(rm+1 − 1)/(r − 1), rm, rm−1(r − 1)/q), where m is any positive integer and q and r = (qd − 1)/(q − 1) are prime powers, and a family of non-embeddable quasi-residual 2−((r+1)(rm+1−1)/(r−1), rm(r+ 1)/2, rm(r− 1)/2) designs, where m is any positive integer and r = 2d− 1, 3 · 2...

Let χ1(n) denote the maximum possible absolute value of an entry of the inverse of an n by n invertible matrix with 0, 1 entries. It is proved that χ1(n) = n 1 2. This solves a problem of Graham and Sloane. Let m(n) denote the maximum possible number m such that given a set of m coins out of a collection of coins of two unknown distinct weights, one can decide if all the coins have the same wei...

Two algorithmic techniques for specifying the existence of a k×k submatrix with elements 0,±1 in a skew and symmetric conference matrix of order n are described. This specification is achieved using an appropriate computer algebra system.

In this paper we completely classify the circulant weighing matrices of weight 16 and odd order. It turns out that the order must be an odd multiple of either 21 or 31. Up to equivalence, there are two distinct matrices in CW (31, 16), one matrix in CW (21, 16) and another one in CW (63, 16) (not obtainable by Kronecker product from CW (21, 16)). The classification uses a multiplier existence t...

Gauss periods taking exactly two values are closely related to two-weight irreducible cyclic codes and strongly regular Cayley graphs. They have been extensively studied in the work of Schmidt andWhite and others. In this paper, we consider the questionofwhenGauss periods take exactly three rational values.Weobtain numerical necessary conditions for Gauss periods to take exactly three rational ...

The class of square (0, 1,−1)-matrices whose rows are nonzero and mutually orthogonal is studied. This class generalizes the classes of Hadamard and Weighing matrices. We prove that if there exists an n by n (0, 1,−1)-matrix whose rows are nonzero, mutually orthogonal and whose first row has no zeros, then n is not of the form pk, 2pk or 3p where p is an odd prime, and k is a positive integer.

Compared to binary self-dual codes, few methods are known to construct ternary self-dual codes. In this paper, a construction method for ternary self-dual codes is presented. Using this method, a number of new extremal ternary self-dual codes are obtained from weighing matrices. In addition, a classification is given for extremal ternary self-dual codes of length 40 constructed from Hadamard ma...

Directed sequences have been recently introduced and used for constructing new orthogonal designs. The construction is achieved by multiplying the length and type of suitable compatible sequences. In this paper we show that near normal sequences of length n = 4m + 1 can be used to construct four directed sequences of lengths 2m+1, 2m+1, 2m, 2m and type (4m+1, 4m+1) = (n, n) with zero NPAF. The ...