Structure and Rank of Cyclic codes over a class of non-chain rings

The rings $Z_{4}+\nu Z_{4}$ have been classified into chain rings and non-chain rings on the basis of the values of $\nu^{2} \in Z_{4}+\nu Z_{4}.$ In this paper, the structure of cyclic codes of arbitrary length over the rings $Z_{4}+\nu Z_{4}$ for those values of $\nu^{2}$ for which these are non-chain rings has been established. A unique form of generators of these codes has also been obtained. Further, rank and cardinality of these codes have been established by finding minimal spanning sets for these codes.Comment: 11 page

Reversible cyclic codes over finite chain rings

In this paper, necessary and sufficient conditions for the reversibility of a cyclic code of arbitrary length over a finite commutative chain ring have been derived. MDS reversible cyclic codes having length p^s over a finite chain ring with nilpotency index 2 have been characterized and a few examples of MDS reversible cyclic codes have been presented. Further, it is shown that the torsion codes of a reversible cyclic code over a finite chain ring are reversible. Also, an example of a non-reversible cyclic code for which all its torsion codes are reversible has been presented to show that the converse of this statement is not true. The cardinality and Hamming distance of a cyclic code over a finite commutative chain ring have also been determined

MDS and MHDR cyclic codes over finite chain rings

In this work, a unique set of generators for a cyclic code over a finite chain ring has been established. The minimal spanning set and rank of the code have also been determined. Further, sufficient as well as necessary conditions for a cyclic code to be an MDS code and for a cyclic code to be an MHDR code have been obtained. Some examples of optimal cyclic codes have also been presented

Positive Values of Non-homogeneous Indefinite Quadratic Forms of Type (2, 4)

This article does not have an abstract

On conjectures of minkowski and woods for n=7

Let Rnbe the n-dimensional Euclidean space with O as the origin. Let ∧ be a lattice of determinant 1 such that there is a sphere |X|<R which contains no point of ∧ other than 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 Rn of radius √n/4 contains a point of ∧. This is known to be true for n≤6. Here we prove a more general conjecture of Woods for n=7 from which this conjecture follows in R7. Together with a result of C.T. McMullen (2005), the long standing conjecture of Minkowski follows for n=7

Estimates on conjectures of Minkowski and woods

Let Rn be the n-dimensional Euclidean space. Let Λ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of Λ other than the origin O and has n linearly independent points of Λ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in Rn of radius √n/4 contains a point of Λ. 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

A unified simple proof of a conjecture of woods for n≤6

Let Rn be the n-dimensional Euclidean space. Let L denote a lattice in Rn of determinant 1 such that there is a sphere centered at the origin O which contains n linearly independent points of L on its boundary but no point of L other than O inside it. A well-known conjecture in the geometry of numbers asserts that any closed sphere in Rn of radius ½√n contains a point of L. This is known to be true for n≤6. Here we give a unified simple proof for n≤6 of the more general conjecture of Woods