No abstract

By the reduction theory of quadratic forms introduced by Korkine and Zolotareff [9], a cartesian coordinate system may be chosen in R n in such a way that L has a basis of the form (A 1 , 0, 0,. .. , 0), (a 2,1 , A 2 , 0,. .. ,

Let R n be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in R n of radius n/4 contains a point of ∧. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il'in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms.

Let q be an odd prime power and p be an odd prime with gcdðp; qÞ ¼ 1: Let order of q modulo p be f ; gcdð pÀ1 f ; qÞ ¼ 1 and q f ¼ 1 þ pl: Here expressions for all the primitive idempotents in the ring R p n ¼ GF ðqÞ½x=ðx p n À 1Þ; for any positive integer n; are obtained in terms of cyclotomic numbers, provided p does not divide l if nX2: The dimension, generating polynomials and minimum distances of minimal cyclic codes of length p n over GF ðqÞ are also discussed. r 2004 Elsevier Inc. All rights reserved.

Let R n be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in R n of radius √ n/2 contains a point of ∧. This is known to be true for n ≤ 8. Recently we gave estimates on a more general conjecture of Woods for n ≥ 9. This lead to an improvement for 9 ≤ n ≤ 22 on estimates of Il'in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms. Here we shall refine our method to obtain improved estimates for Woods Conjecture. These give improved estimates of Minkowski's conjecture for 9 ≤ n ≤ 31.