We study the distribution of spacings between squares in Z/QZ as the number of prime divisors of Q tends to infinity. In [3] Kurlberg and Rudnick proved that the spacing distribution for square free Q is Poissonian, this paper extends the result to arbitrary Q.

We construct a class of codes of length such that the minimum distance outside of a certain subcode is, up to a constant factor, bounded below by the square root of , a well-known property of quadratic residue codes. The construction, using the group algebra of an Abelian group and a special partition or splitting of the group, yields quadratic residue codes, duadic codes, and their generalizat...

It is shown that the extended quadratic residue code of length 20 over GF(7) is a unique self-dual [20, 10, 9] code C such that the lattice obtained from C by Construction A is isomorphic to the 20-dimensional unimodular lattice D 20, up to equivalence. This is done by converting the classification of such self-dual codes to that of skew-Hadamard matrices of order 20.

The largest minimum weight of a self-dual doubly-even binary (n, k, d) code is d = 4bn/24c+ 4. Of such codes with length divisible by 24, the Golay Code is the only (24, 12, 8) code, the Extended Quadratic Residue Code is the only known (48, 24, 12) code, and there is no known (72, 36, 16) code. One may partition the search for a (48, 24, 12) self-dual doubly-even code into 3 cases. A previous ...

1 Quadratic Congruences to Prime Moduli . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Quadratic Congruences to Composite Moduli . . . . . . . . . . . . . . . . . . . . . 5 3 Some Sums of Legendre’s symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Solutions . . . . . . . . . . . . ...

In this paper, a decoding method is proposed for the (41, 21, 9) quadratic residue code. The method is acquired through applying the condition of the modified Gao’s algorithm that is similar to fast transform decoding developed by Shiozaki [8]. The property of the syndrome polynomial is used in our decoding method. The new algorithm for has been verified by a software simulation using C languag...

— For given positive integers a, b, q we investigate the density of solutions (x, y) ∈ Z to congruences ax+ by ≡ 0 mod q.

Wedescribe a polynomial-size conic quadratic reformulation for amachine-job assignment problemwith separable convex cost. Because the conic strengthening is based only on the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation. © 2009 Elsevier B.V. All rights reserved.

Let C be a binary extremal self-dual doubly-even code of length n ≥ 48. If such a code has an automorphism σ of prime order p ≥ 5 then the number of fixed points in the permutation action on the coordinates is bounded by the number of p-cycles. It turns out that large primes p, i.e. n − p small, occur extremely rarely in Aut(C). Examples are the extended quadratic residue codes. We prove that d...

Given an arbitrary n, we consider anisotropic quadratic forms of dimension n over all fields of characteristic = 2 and prove that the height of an n-dimensional excellent form (depending on n only) is the (precise) lower bound of the heights of all forms of dimension n.