A fast algorithm for determining the minimal polynomial and linear complexity of a upn-periodic sequence over a finite field Fq is given.Let p,q,and u be distinct primes,q a primitive root modulo p2,m the smallest positive integer such that qm≡1 mod u,and gcd（m,p（p-1））=1.An algorithm is used to reduce a periodic upn sequence over Fq to several pn-periodic sequences over Fq（ζ）,where ζ is a u-th primitive root of unity,and an algorithm proposed by Xiao et al.is employed to obtain the minimal polynomial of each pn-periodic sequence.
In this paper, we construct some families of strongly regular graphs on finite fields by using unions of cyclotomic classes and index 2 Gauss sums. New infinite families of strongly regular graphs are found.
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.
In this paper, we construct some cyclic division algebras （K/F,σ,γ）. We obtain a necessary and sufficient condition of a non-norm elementγ provided that F = Q and K is a subfield of a cyclotomic field Q（ζpu）, where p is a prime and ζpu is a pu th primitive root of unity. As an application for space time block codes, we also construct cyclic division algebras （K/F,σ, γ）, where F = Q（i）, i = √-1, K is a subfield of Q（ζ4pu） or Q（ζ4pu1 pu2）, and γ = 1＋i. Moreover, we describe all cyclic division algebras （K/F, σ, γ） such that F = Q（i）, K is a subfield of L = Q（ζ4pu1, pu2） and γ= 1 ＋i, where [K： F] = φ（pu1 pu2）/d, d = 2 or 4, φ is the Euler totient function, and p1,p2 ≤ 100 are distinct odd primes.
