On symmetric algorithms for bilinear forms over finite fields

In this paper we study the computation of symmetric systems of bilinear forms over finite fields via symmetric bilinear algorithms. We show that, in general, the symmetric complexity of a system is upper bounded by a constant multiple of the bilinear complexity; we characterize symmetric algorithms in terms of the cosets of a specific cyclic code, and we show that the problem of finding an optimal symmetric algorithm is equivalent to the maximum-likelihood decoding problem for this code.     

On quasi-twisted codes over finite fields

In coding theory, quasi-twisted (QT) codes form an important class of codes which has been extensively studied. We decompose a QT code to a direct sum of component codes – linear codes over rings. Furthermore, given the decomposition of a QT code, we can describe the decomposition of its dual code. We also use the generalized discrete Fourier transform to give the inverse formula for both the nonrepeated-root and repeated-root cases. Then we produce a formula which can be used to construct a QT code given the component codes.     

On nonsystematic perfect codes over finite fields

The nonsystematic perfect q-ary codes over finite field F _q of length n = (q ^m − 1)/(q − 1) are constructed in the case when m ≥ 4 and q ≥ 2 and also when m = 3 and q is not prime. For q ≠ 3, 5, these codes can be constructed by switching seven disjoint components of the Hamming code H _q ⁿ ; and, for q = 3, 5, eight disjoint components.     

On full-rank perfect codes over finite fields

We propose a construction of full-rank q-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by Etzion and Vardy (1994). The properties of the i-components of q-ary Hamming codes are investigated, and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1-perfect codes is generalized to the q-ary case. We propose a generalization of the notion of an i-component of a 1-perfect code and introduce the concept of an (i, σ)-component of a q-ary 1-perfect code. We also present a generalization of the Lindström–Schönheim construction of q-ary 1-perfect codes and provide a lower bound for the number of pairwise distinct q-ary 1-perfect codes of length n.     

On evaluating multivariate polynomials over finite fields

Some evaluation methods of multivariate polynomials over finite fields are described and their multiplicative complexity is discussed.     

On coefficients of polynomials over finite fields

In this paper we study the relation between coefficients of a polynomial over finite field Fq and the moved elements by the mapping that induces the polynomial. The relation is established by a special system of linear equations. Using this relation we give the lower bound on the number of nonzero coefficients of polynomial that depends on the number m of moved elements. Moreover we show that there exist permutation polynomials of special form that achieve this bound when m|q−1. In the other direction, we show that if the number of moved elements is small then there is an recurrence relation among these coefficients. Using these recurrence relations, we improve the lower bound of nonzero coefficients when m?q−1 and . As a byproduct, we show that the moved elements must satisfy certain polynomial equations if the mapping induces a polynomial such that there are only two nonzero coefficients out of 2m consecutive coefficients. Finally we provide an algorithm to compute the coefficients of the polynomial induced by a given mapping with O(q3/2) operations.     

On the equational graphs over finite fields

In this paper, we generalize the notion of functional graph. Specifically, given an equation $E(X,Y)=0$ with variables X and Y over a finite field ${\mathbb{F}}_q$ of odd characteristic, we define a digraph by choosing the elements in ${\mathbb{F}}_q$ as vertices and drawing an edge from x to y if and only if $E(x,y)=0$. We call this graph as equational graph. In this paper, we study the equational graph when choosing $E(X,Y)=(Y^2-f(X))(\lambda Y^2-f(X))$ with $f(X)$ a polynomial over ${\mathbb{F}}_q$ and λ a non-square element in ${\mathbb{F}}_q$. We show that if f is a permutation polynomial over ${\mathbb{F}}_q$, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials f of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.     

On bilinear algorithms over fields of different characteristics

Relationship between bilinear algorithms over fields of different characteristics and over different rings is investigated.     

On the subset sum problem over finite fields

The subset sum problem over finite fields is a well-known NP-complete problem. It arises naturally from decoding generalized Reed–Solomon codes. In this paper, we study the number of solutions of the subset sum problem from a mathematical point of view. In several interesting cases, we obtain explicit or asymptotic formulas for the solution number. As a consequence, we obtain some results on the decoding problem of Reed–Solomon codes.     

On solving sparse algebraic equations over finite fields

A system of algebraic equations over a finite field is called sparse if each equation depends on a small number of variables. In this paper new deterministic algorithms for solving such equations are presented. The mathematical expectation of their running time is estimated. These estimates are at present the best theoretical bounds on the complexity of solving average instances of the above problem. In characteristic 2 the estimates are significantly lower the worst case bounds provided by SAT solvers.     

On the polynomial Ramanujan sums over finite fields

The polynomial Ramanujan sum was first introduced by Carlitz (Duke Math J 14:1105–1120, 1947), and a generalized version by Cohen (Duke Math J 16:85–90, 1949). In this paper, we study the arithmetical and analytic properties of these sums, deriving various fundamental identities, such as Hölder formula, reciprocity formula, orthogonality relation, and Davenport–Hasse type formula. In particular, we show that the special Dirichlet series involving the polynomial Ramanujan sums are, indeed, the entire functions on the whole complex plane, and we also give a square mean values estimation. The main results of this paper are new appearance to us, which indicate the particularity of the polynomial Ramanujan sums.