The Grey-Rankin bound for nonbinary codes

The Grey-Rankin bound for nonbinary codes is obtained. Examples of codes meeting this bound are given. Original Russian Text Ş L.A. Bassalygo, S.M. Dodunekov, V.A. Zinoviev, T. Helleseth, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 3, pp. 37–44. Supported in part by the Russian Foundation for Basic Research, project no. 06-01-00226.     

Open Wireless Networks on University Campuses

Open wireless networks raise privacy issues and entail increased risk of malicious attacks and illegal downloading activities. Such networks are nonetheless attractive—particularly to universities—because they enhance usability and thus expand access to nonsensitive system resources. At universities, such access brings numerous benefits to students, faculty, and the surrounding community alike. Here, the authors describe the challenges of removing individual user authentication requirements at the perimeter of a university network in which mobile device users access system resources over wireless links to the wired infrastructure. The authors discuss how to mitigate the security and privacy risks entailed in an open network of this sort, and also describe how IT departments can vary the network's degree of openness.     

A Generic Construction of Cartesian Authentication Codes

In this paper, a coding-theory construction of Cartesian authentication codes is presented. The construction is a generalization of some known constructions. Within the framework of this generic construction, several classes of authentication codes using certain classes of error-correcting codes are described. The authentication codes presented in this paper are better than known ones with comparable parameters. It is demonstrated that the construction is related to certain combinatorial designs, such as difference matrices and generalized Hadamard matrices     

A New Family of Ternary Almost Perfect Nonlinear Mappings

A mapping f(x) from GF(pⁿ) to GF(pⁿ) is differentially k-uniform if k is the maximum number of solutions x isin GF(pⁿ) of f(x+a) - f(x) = b, where a, b isin GF(pⁿ) and a ne 0. A 2-uniform mapping is called almost perfect nonlinear (APN). This correspondence describes new families of ternary APN mappings over GF(3ⁿ), n>3 odd, of the form f(x) = ux^d + x^{d 2} where d₁ = (3^n-1)/2 - 1 and d₂ = 3ⁿ - 2.     

Some Power Mappings with Low Differential Uniformity

 Differentially uniform power mappings of the form f (x)=x ^d over GF(p ⁿ ) are considered. We construct an infinite family of 2-uniform mappings in the binary case. In the nonbinary case we give two large families of k-uniform mappings with low values of k. We also show how to construct families of sequences from differentially 1-uniform power mappings, which have parameters as good as the best presently known comparable families of sequences. Received: November 4, 1996; revised version: February 14, 1997     

The minimum distance of the duals of binary irreducible cyclic codes

Irreducible cyclic codes have been an interesting subject of study for many years. The weight distribution of some of them have been determined. We determine the minimum distance and certain weights of the duals of binary irreducible cyclic codes. We show that the weight distribution of these codes is determined by the cyclotomic numbers of certain order. As a byproduct, we describe a class of double-error correcting codes.     

Cyclic codes over Z4, locator polynomials, and Newton'sidentities

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock (1972) and Preparata (1968) codes that can be very simply constructed as binary images, under the Gray map, of linear codes over Z₄ that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of Z ₄. Linear codes with length 2^m (m, odd) and size 2(2^m+1-5m-2). The Gray image of the code of length 32 is the best (64, 2³⁷) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over Z₄ that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 2³²) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice     

Upper bound for a hybrid sum over Galois rings with applications toaperiodic correlation of some q-ary sequences

An upper bound for a hybrid exponential sum over Galois rings is derived. This bound is then used to obtain an upper bound for the maximum aperiodic correlation of some sequence families over Galois rings. The bound is of the order of √qlnq where q-1 is the period of the sequences     

The simplex codes and other even-weight binary linear codes for error correction

The probability of correct decoding on the binary-symmetric channel is studied. In particular, a class of codes with the same lengths and dimensions as the linear simplex codes, but with larger probability of correct decoding for all parameters p, 0 < p < 1/2, is given.     

General principles for the algebraic decoding of cyclic codes

This paper provides two theorems for decoding all types of cyclic codes. It is shown that from a polynomial ideal point of view, the decoding problems of cyclic codes are closely related to the monic generators of certain polynomial ideals. This conclusion is also generalized to the decoding problems of algebraic geometry codes.<>