Binary optimal codes often contain optimal or near-optimal subcodes. In this paper we show that this is true for the family of self-dual codes. One approach is to compute the optimum distance profiles (ODPs) of linear codes, which was introduced by Luo, et. al. (2010). One of our main results is the development of general algorithms, called the Chain Algorithms, for finding ODPs of linear codes...

Let G be the automorphism group of the four-dimensional cube, a group of order24 . 4 ! = 384. The binary codes associated with the 32-dimensional permutationrepresentation of G on the edges of the cube are investigated. There are about 400 suchcodes, one of which is a [32, 17, 8] code, having twice as many codewords as the[32, 16, 8] extended quadratic residue code. __________<l...

We consider the problem of finding a trellis for a finear block code that minimizes one or more measures of trellis complexity. The domain of optimization may be different permutations of the same code or different codes with the same parameters. Constraints on trellises, including relationships between the minimal trellis of a code and that of the dual code, are used to derive bounds on comple...

The toric residue is a map depending on n + 1 semi-ample divisors on a complete toric variety of dimension n . It appears in a variety of contexts such as sparse polynomial systems, mirror symmetry, and GKZ hypergeometric functions. In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the as...

The toric residue is a map depending on n + 1 divisors on a complete toric variety of dimension n . It appears in a variety of contexts such as sparse polynomial systems, mirror symmetry, and GKZ hypergeometric functions. In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated po...

In this paper quadratic residue codes over the ring Fp + vFp are introduced in terms of their idempotent generators. The structure of these codes is studied and it is observed that these codes share similar properties with quadratic residue codes over finite fields. For the case p = 2, Euclidean and Hermitian self-dual families of codes as extended quadratic residue codes are considered and two...

Difference Systems of Sets (DSS) are combinatorial configurations that arise in connection with code synchronization. A method for the construction of DSS from partitions of cyclic difference sets was introduced in [6] and applied to cyclic difference sets (n, (n−1)/2, (n−3)/4) of Paley type, where n ≡ 3 (mod 4) is a prime number. This paper develops similar constructions for prime numbers n ≡ ...

In this paper we present a family of ternary quasi-perfect BCH codes. These codes are of minimum distance 5 and covering radius 3. The first member of this family is the ternary quadratic-residue code of length 13.

Let n ∈ N and let A ⊆ Z/nZ be such that A does not contain 0 and it is non–empty. Generalizing a well known constant, EA(n) is defined to be the least t ∈ N such that for all sequences (x1, . . . , xt) ∈ Z, there exist indices j1, . . . , jn ∈ N, 1 ≤ j1 < · · · < jn ≤ t, and (θ1, · · · ,θn) ∈ A with ∑n i=1 θixji ≡ 0 (mod n). Similarly, for any such set A, we define the Davenport Constant of Z/n...

In this paper, we study explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic non-residue and the least prime in an arithmetic progression. We also refine the classical conditional bounds of Littlewood for L-functions at s = 1. In part...