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In this article, we establish a sufficient condition for the existence of a primitive element $\alpha \in {\mathbb{F}_{q^n}}$ such that the element $\alpha+\alpha^{-1}$ is also a primitive element of ${\mathbb{F}_{q^n}},$ and $Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(\alpha)=a$ for any prescribed $a \in \mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field $\mathbb{F}_{q^n}~ (n \geq5),$ contains such primitive elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $(q,n)$ for $n\geq5.$, Comment: 19 pages, 1 table

In this article, we establish a sufficient condition for the existence of a primitive element Fq such that for any matrix (abc0de)M23(Fq) of rank 2, the element (a2+b+c)/(d+e) is a primitive element of Fq, where q=2k for some positive integer k. We also give a sufficient condition for the existence of a primitive normal element Fqn over Fq such that (a2+b+c)/(d+e) is a primitive element of Fqn for every matrix (abc0de)M23(Fqn) of rank 2.

Let α be a primitive element of a finite field Fr, where r=qm1m2 and gcd⁡(m1,m2)=d, so α1=αr−1qm1−1 and α2=αr−1qm2−1 are primitive elements of Fqm1 and Fqm2, respectively. Let e be a positive integer such that gcd⁡(e,qm2−1qd−1)=1, Fqm2=Fq(α2e), and α1 and α2e are not conjugates over Fq. We define a cyclic code C={c(a,b):a∈Fqm1,b∈Fqm2}, c(a,b)=(T1(aα1i)+T2(bα2ei))i=0n−1, where Ti denotes the trace function from Fqmi to Fq for i=1,2. In this paper, we use Gauss sums to investigate the weight distribution of C, which generalizes the results of C. Li and Q. Yue in [13,14]. Furthermore, we explicitly determine the weight distribution of C if d=1,2. Moreover, we prove it is optimal three-weight achieving the Griesmer bound if d=1 and gcd⁡(m2−em1,q−1)=1.

We prove that any finite semifield of dimension four over its center F q is right and left primitive, provided that q is odd and sufficiently large.

For two positive integers m and k, let C e be a class of cyclic code of length 2 m - 1 over F 2 with three nonzeros γ - 1 , γ - ( 2 k + 1 ) and γ - ( 2 e k + 1 ) for e = 2 or 3, where γ is a primitive element of F 2 m . When m gcd ? ( m , k ) is odd, Kasami in 1971 determined the weight distributions of cyclic codes C 2 and C 3 , which is the same as that of the dual of the triple error-correcting BCH code. This paper obtains the weight distributions of C 2 and C 3 for the case of m gcd ? ( m , k ) being even. A certain system of algebraic equations over finite fields is solved.A connection between the "type" and rank of the quadratic forms in certain case is applied.The weight distribution of a class of binary cyclic codes is determined by using the connection.We determine another class of binary cyclic codes by using the Pless power moment identities

Let q be a prime power, m>=2 an integer and A=(abcd)@?GL"2(F"q), where A (1101) if q=2 and m is odd. We prove an extension of the primitive normal basis theorem and its strong version. Namely, we show that, except for an explicit small list of genuine exceptions, for every q, m and A, there exists some primitive [email protected]?F"q"^"m such that both x and (ax+b)/(cx+d) produce a normal basis of F"q"^"m over F"q.

An element α ? F q n is normal over F q if { α , α q , ? , α q n - 1 } is a basis for F q n over F q . It is well known that α ? F q n is normal over F q if and only if the polynomials g α ( x ) = α x n - 1 + α q x n - 2 + ? + α q n - 2 x + α q n - 1 and x n - 1 are relatively prime over F q n , that is, the degree of their greatest common divisor in F q n x is 0. An element α ? F q n is k-normal over F q if the greatest common divisor of the polynomials g α ( x ) and x n - 1 in F q n x has degree k; so an element which is normal in the usual sense is 0-normal. In this paper, we introduce and characterize k-normal elements, establish a formula and numerical bounds for the number of k-normal elements and prove an existence result for primitive 1-normal elements.

The proof of the theorem that there exists a primitive polynomial of degree n ⩾ 3 over F q with arbitrary trace a and arbitrary primitive norm b is completed by establishing the result when n = 3 or 4 and a = 0 . The only exceptions occur when n = 3 , a = 0 and q = 4 or 7.

We consider the ring R p n q = GF ( l ) [ x ] / ( x p n q − 1 ) , where p , q , l are distinct odd primes, l is a primitive root both modulo p n and q . Explicit expressions for all the ( d + 1 ) n + 2 primitive idempotents are obtained, d = gcd ( ϕ ( p n ) , ϕ ( q ) ) , p ∤ ( q − 1 ) . The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length p n q over GF ( l ) are also discussed.

Let q be a power of 2, n be a positive integer, and let F q n be the finite field with q n elements. In this paper, we consider the existence of some specific elements in F q n . The main results obtained in this paper are listed as follows: (1) There is an element ξ in F q n such that both ξ and ξ + ξ − 1 are primitive elements of F q n if q = 2 s , and n is an odd number no less than 13 and s > 4 . (2) For q = 2 s , and any odd n, there is an element ξ in F q n such that ξ is a primitive normal element and ξ + ξ − 1 is a primitive element of F q n if either n | ( q − 1 ) , and n ⩾ 33 , or n ∤ ( q − 1 ) , and n ⩾ 30 , s ⩾ 6 .

Based on a sufficient condition proposed by Hollmann and Xiang for constructing triple-error-correcting codes, the minimum distance of a binary cyclic code C 1 , 3 , 13 with three zeros α, α 3 , and α 13 of length 2 m − 1 and the weight divisibility of its dual code are studied, where m ⩾ 5 is odd and α is a primitive element of the finite field F 2 m . The code C 1 , 3 , 13 is proven to have the same weight distribution as the binary triple-error-correcting primitive BCH code C 1 , 3 , 5 of the same length.

For an odd prime p and two positive integers n>=3 and k with ngcd(n,k) being odd, the paper determines the weight distribution of a class of p-ary cyclic codes C over F"p with nonzeros @a^-^1, @a^-^(^p^^^k^+^1^) and @a^-^(^p^^^3^^^k^+^1^), where @a is a primitive element of F"p"^"n.

In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n⩾15 be a positive integer and q a prime power. We prove that for any a∈Fq and any 1⩽m

In this paper, we prove that for any given n>=2, there exists a constant C(n) such that for any prime power q>C(n), there exists a primitive normal polynomial of degree n over F"q with the last @[email protected]? coefficients prescribed, where the last coefficient is a primitive element. Furthermore, the number of prescribed coefficients increases from @[email protected]? to @[email protected]? when the coefficients are specified as (0,...,0,b) with b any primitive element. This result is a complement to the existence of a primitive normal polynomial with the first @[email protected]? coefficients prescribed which was proved in [S.Q. Fan, W.B. Han, K.Q. Feng, Primitive normal polynomials with multiple coefficients prescribed: An asymptotic result, Finite Fields Appl. 13 (2007) 1029-1044]. The outline of this paper is similar to the above reference with the following two different treatments. On one hand, we use 1/x instead of x in the problem reduction step and as a consequence use the hybrid kloostermann sums instead of hybrid weil sums over Galois rings. On the other hand, the estimates are slightly more complicated and the results in some special cases are better than those in [S.Q. Fan, W.B. Han, K.Q. Feng, Primitive normal polynomials with multiple coefficients prescribed: An asymptotic result, Finite Fields Appl. 13 (2007) 1029-1044].

This paper studies the distinctness problem of the reductions modulo 2 of maximal length sequences over Z/(pq), where p and q are two different odd primes with p1, it is proved that if there exist a nonnegative integer S and a primitive element @x in Z/(pq) such that x^[email protected]=0(modf(x),pq), and either (q-1) is not divisible by (p-1) or 2(p-1) divides (q-1), then [email protected][email protected]?(mod2) if and only if [email protected][email protected]?. The existence of S and @x is completely determined by p, q and degf(x). Secondly, for the case of degf(x)=1, it is proved that if gcd(p-1,q-1)=2 and (p-1)/ord"p(2) is congruent to (q-1)/ord"q(2) modulo 2, then [email protected][email protected]?(mod2) if and only if [email protected][email protected]?.

In this paper, we prove that for any given n>=2, there exists a constant C(n) such that for any prime power q>C(n), there exists a primitive normal polynomial of degree n over F"q with the first @[email protected]? coefficients prescribed, where the first coefficient is nonzero. This result strengthens the asymptotic result of the existence of primitive polynomials with the first @[email protected]? coefficients prescribed [S.Q. Fan, W.B. Han, p-Adic formal series and Cohen's problem, Glasg. Math. J. 46 (2004) 47-61] in two aspects. One is that we discuss in this paper not only the primitivity but also the normality. Another is that the number of the prescribed coefficients increases from @[email protected]? to @[email protected]?. The estimates of character sums over Galois rings, the p-adic method introduced by the first two authors, and the computation technique used in [S.Q. Fan, W.B. Han, Primitive polynomial with three coefficients prescribed, Finite Fields Appl. 10 (2004) 506-521; D. Mills, Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl. 4 (2004) 1-22] are the main tools to get the above result.

Explicit expressions for all the 3n+2 primitive idempotents in the ring R"p"^"n"q=GF(@?)[x]/(x^p^^^n^q-1), where p,q,@? are distinct odd primes, @? is a primitive root modulo p^n and q both, gcd(@f(p^n)2,@f(q)2)=1, are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length p^nq over GF(@?) are also discussed.

We continue to study the existence of (norm- and) trace-compatible sequences of primitive normal bases for prime power extensions of finite fields, introduced by the author in Hachenberger (Finite Fields Appl. 5 (1999) 378–385; in: D. Jungnickel, H. Niederreiter (Eds.), Proceedings of the Fifth International Conference on Finite Fields and Applications, Augsburg, August 1999, Springer, Heidelberg, 2001, pp. 208–223), and improve on some aspects of these papers.

An asymptotic formula is obtained for the number of representations of an element of a finite field as a weighted sum of two prescribed powers of primitive elements. This generalises previous work on sums of primitive elements, including that relating to some conjectures of Golomb.

The paper is devoted to some results concerning the constructive theory of the synthesis of irreducible polynomials over Galois fields GF(q), q=2s. New methods for the construction of irreducible polynomials of higher degree over GF(q) from a given one are worked out. The complexity of calculations does not exceed O(n3) single operations, where n denotes the degree of the given irreducible polynomial. Furthermore, a recurrent method for constructing irreducible (including self-reciprocal) polynomials over finite fields of even characteristic is proposed.

Explicit expressions for the (n+ 1) primitive idempotents inFG(the group algebra of the cyclic groupGof orderpn(podd prime,n> 1) over the finite fieldFof prime power orderqwhereqis a primitive root modulopn) are obtained. The minimum distance, the dimension, and the generating polynomial of the (n+ 1) minimum codes generated by the primitive idempotents are also obtained.

Given an extension E/F of Galois fields and an intermediate field K, we consider the problem whether the (E, K)-trace of a primitive F-normal element of E can be a prescribed F-normal element of K. An interesting application is the existence of trace-compatible sequences of primitive F-normal elements for certain towers of Galois fields. In this respect, particular emphasis is laid on extensions having prime power degree.

In numerous applications involving finite fields, we often need high-order elements. Ideally we should be able to obtain a primitive element for any finite field in reasonable time. However, if the prime factorization of the group order is unknown, we do not know how to achieve the goal. We thus turn our attentions to a less ambitious problem: constructing an element of provably high order. In this paper, we survey various algorithms that find an element of high order for general or special finite fields.

Let @w be a primitive element of GF(2^n), where n=0(mod4). Let d=(2^2^k+2^s^+^1-2^k^+^1-1)/(2^s-1), where n=2k, and s is such that 2s divides k. We prove that the binary m-sequences s(t)=tr(@w^t) and s(dt) have a four-level cross-correlation function and give the distribution of the values.

Binary m-sequences are widely applied in navigation, radar, and communication systems because of their nice autocorrelation and cross-correlation properties. In this paper, we consider the cross-correlation between a binary m-sequence of length 2m−1 and a decimation of that sequence by an integer t. We will be interested in the number of values attained by such cross-correlations. As is well known, this number equals the number of nonzero weights in the dual of the binary cyclic code C1,t of length 2m−1 with defining zeros α and αt, where α is a primitive element in GF(2m). There are many pairs (m, t) for which C⊥1,t is known or conjectured to have only few nonzero weights. The three-weight examples include the following cases:We present a method of proving many of these known or conjectured results, including all of the above cases, in a unified way.

Let Fq denote the finite field of order q, a power of a prime p, and n be a positive integer. We resolve completely the question of whether there exists a primitive element of Fqn which is such that it and its reciprocal both have zero trace over Fq. Trivially, there is no such element when n

The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given (a"1,a"2,a"3)@?F"q^3, there exists a primitive polynomial f(x)=x^[email protected]"1x^n^-^1+...+(-1)^[email protected]"n over F"q of degree n with the first three coefficients @s"1,@s"2,@s"3 prescribed as a"1,a"2,a"3 when n>=8. But the methods in Fan and Han (in press) are not effective for the case of n=7. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the n=7 case for finite fields of characteristic at least 5. In this paper, we deal with the remaining cases and prove that there exists a primitive polynomial of degree 7 over F"q with the first three coefficient prescribed where the characteristic of F"q is 2 or 3.

Progress over the past decade is surveyed concerning explicit existence and construction theorems on irreducible, primitive and normal polynomials.

Let N be a product of distinct prime numbers and Z/(N) be the integer residue ring modulo N. In this paper, a primitive polynomial f(x) over Z/(N) such that f(x) divides xs−c for some positive integer s and some primitive element c in Z/(N) is called a typical primitive polynomial. Recently typical primitive polynomials over Z/(N) were shown to be very useful, but the existence of typical primitive polynomials has not been fully studied. In this paper, for any integer m⩾1, a necessary and sufficient condition for the existence of typical primitive polynomials of degree m over Z/(N) is proved.

In this work, we prove the existence of left and right primitive elements in semifields of prime degree over their centre Fq, for q large enough.

Let F"q be a finite field with q=p^k elements. We prove that for any given n>=7, and any elements a,b@?F"q, a 0, there exists a primitive normal polynomial f(x) of degree n, f(x)=x^n-@s"1x^n^-^1+...+(-1)^n@s"n, with the first two coefficients @s"1, @s"2 prescribed as a, b, respectively. This result strengthens the results of the existence of primitive polynomials with two coefficients prescribed [S.D. Cohen, D. Mills, Primitive polynomials with the first and second coefficients prescribed, Finite Fields Appl. 9 (2003) 334-350; W.B. Han, The coefficients of primitive polynomials over finite fields, Math. Comp. 65 (1996) 331-340; W.B. Han, On two exponential sums and their applications, Finite Fields Appl. 3 (1997) 115-130] and the existence of primitive normal bases with prescribed trace [S.D. Cohen, D. Hachenberger, Primitive normal bases with prescribed trace, Appl. Algebra Engrg. Comm. Comput. 9 (1999) 383-403]. In order to use the p-adic method which proposed by the first two authors, we first lift the definition of both primitive and normal from finite fields to Galois rings. Then we discuss the existence of lifted primitive normal polynomials over Galois rings and finally establish the existence of primitive normal polynomials with the first two coefficients prescribed with the help of character sums over Galois rings and Cohen's various sieve techniques. In order to make this paper more succinct, we deal with the case n=4,5,6 in another paper since the computation is more miscellaneous and complicated.