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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.

In the original view-obstruction problem congruent closed, centrally symmetric convex bodies centred at the points of the set are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size and this value is known for balls in dimensions n = 2,3 and for symmetrically placed cubes in dimensions n = 2, 3, 4In order to explain fully the distinction between rational and irrational rays in the original problem, we extend consideration to the blocking of subspaces of all dimensions. In order to appreciate the special properties of balls and cubes, we give a discussion of the convex body with respect to reflection symmetry, lower dimensional sections, and duality. We introduce topological considerations to help understand when the critical parameter of the theory is an attained maximum and we add substantially to the list of known values of this parameter. In particular, when the dimension is n = 2 our dual body considerations furnish a complete solution to the view-obstruction problem

A well-known theorem of Minkowski on the product of two linear forms states that ifare two linear forms with real coefficients and determinant Δ = |αδ − βγ| ≠ 0, then given any real numbers c1, c2 we can find integers x, y such that