Showing papers in "Finite Fields and Their Applications in 2002"

TL;DR: Using Fourier transforms and Gauss sums, necessary and sufficient numerical conditions for an irreducible cyclic code to have at most two weights are obtained and this gives a unified explanation for all two-weight irreduceable cyclic codes and allows a conjecturally complete classification.

TL;DR: In this paper, a necessary condition on the algebraic normal form of correlation immune functions attaining the maximum possible nonlinearity was derived from a general property of Fourier transform.

TL;DR: The notion of an order domain is generalized in this paper, where the behavior of order domains by taking a subalgebra, the extension of scalars, and the tensor product is studied.

TL;DR: In this paper, the a-invariant and the defining ideal for the set S"K of rational points of the Segre variety over a finite field K is given and the dimension as well as the minimum distance of a Reed-Muller-type linear code defined over S" K are also determined.

TL;DR: The first extremal self-dual 92, 46, 16] binary code was constructed by as discussed by the authors for the Euclidean scalar product with Lee weights multiple of 4.

TL;DR: A new method is given that creates and solves Diophantine equations over GF(p) in order to find suitable generator matrices for self-dual codes and it is shown that under the necessary conditions these methods can be applied as well to small and large fields.

TL;DR: In this paper, the number of rational points of a class of Artin-Schreier curves was determined by using recent results concerning evaluations of some exponential sums, and infinitely many new examples of maximal and minimal plane curves in the context of the Hasse-Weil bound were given.

TL;DR: An efficient family of linear feedback shift registers (LFSR's) with maximal period is introduced and is suitable for implementation in software, thus providing a solution to a recent challenge 8.

TL;DR: In this article, the completeness of the coset of a subgroup of a non-singular cubic curve is shown to be an arc in the Galois plane of order q, except for the case j = 0, where j is the invariant of the underlying cubic curve.

TL;DR: The inversive congruential method is an attractive alternative to the classical linear congruent method for pseudorandom number generation as mentioned in this paper, and it has been shown that, on average, much stronger results than those known for ''individual'' sequences can be obtained.

TL;DR: In this article, it was shown that a certain fivefold is modular in the sense that for every odd p, the number of its points over Fp is predicted explicitly by the pth coefficient of the Fourier expansion of the weight 6 modular form?12(2z).

TL;DR: Niederreiter as discussed by the authors proved that Khintchine?s Theorem has an analogue in the field of formal Laurent series and showed that in case of only one inequality this result can also be obtained by continued fraction theory.

TL;DR: Borders for the number of strings S of length n with k-error linear complexity Ln,k(S)=c or Ln(S)?c for a given c are established and exact formulas are determined under certain conditions on n, k, and c.

TL;DR: In this article, the authors extend the method of the previous paper and prove an asymptotic formula for the number of permutations for which the associated permutation polynomial has d coefficients in specified fixed positions equal to 0.

TL;DR: In this article, the authors studied the constructive theory of the synthesis of irreducible polynomials over Galois fields GF(q), q = 2s, with complexity of O(n3) single operations, where n denotes the degree of the given polynomial.

TL;DR: In the first part of the paper as discussed by the authors, certain incomplete character sums over a finite field Fpr are considered which in the case of finite prime fields Fp are of the form?A+N?1n=A?(g(n))?(f(n)), where A and N are integers with 1?N

TL;DR: In this paper, the authors improved lower asymptotic bounds of the number of rational points of smooth algebraic curves over F3 and F5 by using ramified Hilbert Class Field Towers.

TL;DR: Garcia, Stichtenoth, and Thomas as discussed by the authors showed that their construction does not work over prime fields, and showed that for every finite field E that is not a prime field, an explicit sequence of absolutely irreducible smooth projective curves Cn over E with genus tending to infinity and with #Cn(E)/genus(Cn) tending to a positive limit can not be constructed.

TL;DR: The authors show that a simple modification of their method can be used to obtain the matrices which are given by the “classical,” more involved construction going back to Berman, and a wealth of monomially inequivalent examples, namely matrices for many different q-ranks.

TL;DR: In this article, a strongly primitive polynomial of degree n over Z/(2^e) was shown to be injective over the Galois rings, where the compressing mapping is injective if and only if @f(a"0,...,a"e"-"1) mod 2.

TL;DR: In this paper, it was shown that for p prime and sufficiently large subsets of Zp free of solutions of the equation x+y=z (that is, free of Schur triples) satisfies 2?(p?2)/3?p?1)(1+O(2??1p))??SFZp]??2p/2.2p, where?1 and?2 are positive absolute constants.

TL;DR: In this paper, the number of permutation polynomials in F"q[x] of degree [email protected]?q-2 was derived in terms of the permanent of a Vandermonde matrix whose entries are the primitive pth roots of unity.

TL;DR: In this article, it was shown that the first slope of the Newton polygon is always lying above the Hodge polygon of the polynomial L(f mod p;T ).

TL;DR: In this paper, an asymptotic formula for the number of representations of an element of a finite field as a weighted sum of two prescribed powers of primitive elements is given.

TL;DR: For any finite field k, we count the number of orbits of galois invariant n-sets of P1(k?) under the action of PGL2(k) as mentioned in this paper.

TL;DR: In this article, the problem of factoring a polynomial P?FPX] over a finite extension of FP has been studied, and algorithms to solve this problem have been given.

TL;DR: In this paper, it was shown that the distribution of Diffie?Hellman pairs (x, gx) is close to uniform in the Cartesian product Zt×Fp, where x runs through the residue ring Z modulo.

TL;DR: In this paper, a continued fraction expansion for power series over a finite field, with all the partial quotients of degree one, which are non-quadratic algebraic elements over the field of rational functions, is presented.

TL;DR: In this paper, the known permutation behaviour of the Dickson polynomials of the second kind in characteristic 3 is expanded and simplified, and the permutation behavior of the first-order polynomial of characteristic 3 has been studied.