49 research outputs found
On the accuracy of the binomial approximation to the distance distribution of codes
The binomial distribution is a well-known approximation to the distance spectra of many classes of codes. We derive a lower estimate for the deviation from the binomial approximatio
An improved upper bound on the minimum distance of doubly-even self-dual codes
We derive a new upper bound on the minimum distance d of doubly-even self-dual codes of length n. Asymptotically, for n growing, it gives limnââ sup d/n <=(5-5^3/4)/10 <0.165630, thus improving on the Mallows-Odlyzko-Sloane bound of 1/6 and our recent bound of 0.16631
On spectra of BCH codes
Derives an estimate for the error term in the binomial approximation of spectra of BCH codes. This estimate asymptotically improves on the bounds by Sidelnikov (1971), Kasami et al. (1985), and Sole (1990)
On the distance distribution of duals of BCH codes
We derive upper bounds on the components of the distance distribution of duals of BCH codes. Roughly speaking, these bounds show that the distance distribution can be upper-bounded by the corresponding normal distribution. To derive the bounds we use the linear programming approach along with some estimates on the magnitude of Krawtchouk polynomials of fixed degree in a vicinity of q/
Estimates for the range of binomiality in codes' spectra
We derive new estimates for the range of binomiality in a codeâs spectra, where the distance distribution of a code is upperbounded by the corresponding normalized binomial distribution. The estimates
depend on the codeâs dual distance
Asymptotically Good Quantum Codes
Using algebraic geometry codes we give a polynomial construction of quantum
codes with asymptotically non-zero rate and relative distance.Comment: 15 pages, 1 figur
New Upper Bounds on Codes via Association Schemes and Linear Programming
Let A(n, d) denote the maximum number of codewords in a binary code of length n and minimum Hamming distance d. Upper and lower bounds on A(n, d) have been a subject for extensive research. In this paper we examine upper bounds on A(n, d) as a special case of bounds on the size of subsets in metric association scheme. We will first obtain general bounds on the size of such subsets, apply these bounds to the binary Hamming scheme, and use linear programming to further improve the bounds. We show that the sphere packing bound and the Johnson bound as well as other bounds are special cases of one of the bounds obtained from association schemes. Specific bounds on A(n, d) as well as on the sizes of constant weight codes are also discussed
On Perfect Weighted Coverings with Small Radius
We extend the results of our previous paper [8] to the nonlinear case: The Lloyd polynomial of the covering has at least R distinct roots among 1, ... , n, where R is the covering radius. We investigate PWC with diameter 1, finding a partial characterization. We complete an investigation begun in [8] on linear PMC with distance 1 and diameter 2