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 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.

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Recent studies involve various approaches to establish a generating set for cyclic codes of arbitrary length over the class of Galois rings. One such approach involves the use of polynomials with minimal degree corresponding to specific subsets of the code, defined progressively. In this paper, we extend this approach to obtain a set of generators of cyclic codes over finite chain rings. Further, we observe that this set acts as a minimal strong Gröbner basis(MSGB) for the code.