GAUSS PERIODS AND CYCLIC CODES FROM CYCLOTOMIC SEQUENCES OF SMALL ORDERS

Let p =ef +1 be an odd prime with positive integers e and f. In this paper, we calculate the values of Gauss periods of order e =3, 4, 6 over a finite field GF(q), where q is a prime with q≠p. As applications, several cyclotomic sequences of order e =3, 4, 6 are employed to construct a number of classes of cyclic codes over GF(q) with prime length. Under certain conditions, the linear complexity and reciprocal minimal polynomials of cyclotomic sequences are calculated, and the lower bounds on the minimum distances of these cyclic codes are obtained.

A Small Correction to a Paper of Vandermonde

In this paper using elementary Galois Theory, we give a detailed explanation of the calculation of the radical expression for which was first discussed by Vandermonde decades before Galois and we point out and correct a minor correction in his work which was also observed by Lagrange.

On the Linear Complexity of New Generalized Cyclotomic Binary Sequences of Order Two and Period pqr

Periodic sequences over finite fields, constructed by classical cyclotomic classes and generalized cyclotomic classes, have good pseudorandom properties. The linear complexity of a period sequence plays a fundamental role in the randomness of sequences. Let p, q, and r be distinct odd primes with gcd（p-1, q-1 ）=gcd（p- 1, r-1）=gcd（q-1, r-1）=2. In this paper, a new class of generalized cyclotomic sequence with respect to pqr over GF（2） is constructed by finding a special characteristic set. In addition, we determine its linear complexity using cyclotomic theory. Our results show that these sequences have high linear complexity, which means they can resist linear attacks.

Autocorrelation Values of New Generalized Cyclotomic Sequences of Order Two and Length pq

Pseudo-random sequences are used extensively for their high speed and security level and less errors. As a branch, the cyclotomic sequences and the generalized ones are studied widely because of their simple mathematical structures and excellent pseudo-random properties. In 1998, Ding and Helleseth introduced a new generalized cyclotomy which includes the classical cyclotomy as a special case. In this paper, based on the generalized cyclotomy, new generalized cyclotomic sequences with order two and length pq are constructed. An equivalent definition of the sequences is deduced so that the autocorrelation values of these sequences can be determined conveniently. The construction contributes to the understanding of the periodic autocorrelation structure of cyclotomically-constructed binary sequences, and the autocorrelation function takes on only a few values.

Linear complexity of quaternary sequences with odd period and low autocorrelation

Equivalence between two classes of quaternary sequences with odd period and best known autocorrelation are proved. A lower bound on the linear complexity of these sequences is presented. It is shown that the quaternary sequences have large linear complexity to resist Reeds and Sloane algorithm attack effectively.

Non Existence of Some Generalized Bent Functions

Several new results on the non-existence of some generalized bent functions are proved by using properties of the decomposition law of primes in cyclotomic fields and properties of the solutions of some special Diophantine equations.

Some Notes on Prime-Square Sequences

The well-known binary Legendre sequences possess good autocorrelation functions and high linear complexity, and are just special cases of much larger families of cyclotomic sequences. Prime-square sequences are the generalization of these Legendre sequences, but the ratio of the linear complexity to the least period of these sequences approximates to zero if the prime is infinite. However, a relatively straightforward modification can radically improve this situation. The structure and properties, including linear complexity, minimal polynomial, and autocorrelation function, of these modified prime-square sequences are investigated. The hardware implementation is also considered.

Generalized Cyclotomic Mappings: Switching Between Polynomial, Cyclotomic, and Wreath Product Form

This paper is concerned with so-called index d generalized cyclotomic mappings of a finite field F_(q), which are functions F_(q)→F_(q) that agree with a suitable monomial function x↦axr on each coset of the index d subgroup of F_(q)^(*). We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index d generalized cyclotomic permutations of F_(q) and pertain to cycle structures, the classification of (q−1)-cycles and involutions, as well as inversion.