Tor Helleseth

Some results about the cross-correlation function between two maximal linear sequences

Let {a1} and {ad1} be two maximal linear sequences of period pn − 1. The cross-correlation function is defined by Cd(t) = for t = 0, t…pn − 2, where ζ = exp(2π 1p). We find some new general results about Cd(t). We also determine the values and the number of occurences of each value of Cd(t) for several new values of d.

An upper bound for some exponential sums over Galois rings and applications

We present an analog of the well-known Weil-Carlitz-Uchiyama (1948, 1957) upper bound for exponential sums over finite fields for exponential sums over Galois rings. Some examples are given where the bound is tight. The bound has immediate application to the design of large families of phase-shift-keying sequences having low correlation and an alphabet of size p/sup e/. p, prime, e/spl ges/2. Some new constructions of eight-phase sequences are provided. >

Almost difference sets and their sequences with optimal autocorrelation

Almost difference sets have interesting applications in cryptography and coding theory. We give a well-rounded treatment of known families of almost difference sets, establish relations between some difference sets and some almost difference sets, and determine the numerical multiplier group of some families of almost difference sets. We also construct six new classes of almost difference sets, and four classes of binary sequences of period n/spl equiv/0 (mod 4) with optimal autocorrelation. We have also obtained two classes of relative difference sets and four classes of divisible difference sets (DDSs). We also point out that a result due to Jungnickel (1982) can be used to construct almost difference sets and sequences of period 4l with optimal autocorrelation.

The weight distribution of irreducible cyclic codes with block lengths n1((q1−1)N)

Abstract We study the weight distribution of irreducible cyclic ( n , k ) codeswith block lengths n = n 1 (( q 1 − 1)/ N ), where N | q − 1, gcd ( n 1 , N ) = 1, and gcd ( l , N ) = 1. We present the weight enumerator polynomial, A ( z ), when k = n 1 l , k = ( n 1 − 1) l , and k = 2 l . We also show how to find A ( z ) in general by studying the generator matrix of an ( n 1 , m ) linear code, V ∗ d over GF ( q d ) where d = gcd ( ord n 1 ( q ), l ). Specifically we study A ( z ) when V ∗ d is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers A μ which completely determine the weight distributionof any irreducible cyclic ( n 1 (2 1 − 1), k ) code over GF(2) for all n 1 ⩽ 17.

Large families of quaternary sequences with low correlation

A family of quaternary (Z/sub 4/-alphabet) sequences of length L=2/sup r/-1, size M/spl ges/L/sup 2/+3L+2, and maximum nontrivial correlation parameter C/sub max//spl les/2/spl radic/(L+1)+1 is presented. The sequence family always contains the four-phase family /spl Ascr/. When r is odd, it includes the family of binary Gold sequences. The sequence family is easily generated using two shift registers, one binary, the other quaternary. The distribution of correlation values is provided. The construction can be extended to produce a chain of sequence families, with each family in the chain containing the preceding family. This gives the design flexibility with respect to the number of intermittent users that can be supported, in a code-division multiple-access cellular radio system. When r is odd, the sequence families in the chain correspond to shortened Z/sub 4/-linear versions of the Delsarte-Goethals codes.

Monomial and quadratic bent functions over the finite fields of odd characteristic

Considered are p-ary bent functions having the form f(x)=Tr/sub n/(/spl sigma//sub i=0//sup s/a/sub i/x/sup di/). A new class of ternary monomial regular bent function with the Dillon exponent is discovered. The existence of Dillon bent functions in the general case is an open problem of deciding whether a certain Kloosterman sum can take on the value -1. Also described is the general Gold-like form of a bent function that covers all the previously known monomial quadratic cases. The (weak) regularity of the new as well as of known monomial bent functions is discussed and the first example of a not weakly regular bent function is given. Finally, some criteria for an arbitrary quadratic function to be bent are proven.

Generalized Hamming weights of linear codes

The generalized Hamming weight, d/sub r/(C), of a binary linear code C is the size of the smallest support of any r-dimensional subcode of C. The parameter d/sub r/(C) determines the codes performance on the wire-tap channel of Type II. Bounds on d/sub r/(C), and in some cases exact expressions, are derived. In particular, a generalized Griesmer bound for d/sub r/(C) is presented and examples are given of codes meeting this bound with equality. >

Several classes of binary sequences with three-level autocorrelation

In this correspondence we describe several classes of binary sequences with three-level autocorrelation. Those classes of binary sequences are based on cyclic almost difference sets. Some classes of binary sequences have optimum autocorrelation.

Niho type cross-correlation functions via dickson polynomials and Kloosterman sums

Suppose that n=2k is even. We study the cross-correlation function between two m-sequences for Niho type decimations d=(2/sup k/-1)s+1. We develop a new technique to study the value distribution of these cross-correlation functions, which makes use of Dickson polynomials. As a first application, we derive here the distribution of the six-valued cross-correlation function for s=3 and odd k, up to a term which depends on Kloosterman sums. In addition, applying simpler methods, we prove a theorem providing Niho type decimations with four-valued cross-correlation functions and their distribution. We conjecture that the latter result actually covers all such decimations.

New families of almost perfect nonlinear power mappings

A power mapping f(x)=x/sup d/ over GF(p/sup n/) is said to be differentially k-uniform if k is the maximum number of solutions x/spl isin/GF(p/sup n/) of f(x+a)-f(x)=b where a, b/spl isin/GF(p/sup n/) and a/spl ne/0. A 2-uniform mapping is called almost perfect nonlinear (APN). We construct several new infinite families of nonbinary APN power mappings.