On the number of roots of some linearized polynomials

Journal of mechanics of continua and mathematical sciences, 2019

In this work we try to introduce the concept of codes of polynomial type and polynomial codes that are built over the ring A[X]/A[X]f(X).It should be noted that for particular cases of f we will find some classic codes for example cyclic codes, constacyclic codes, So the study of these codes is a generalization of linear codes.

arXiv (Cornell University), 2015

We estimate the average cardinality V(A) of the value set of a general family A of monic univariate polynomials of degree d with coefficients in the finite field Fq. We establish conditions on the family A under which V(A) = µ d q + O(q 1/2), where µ d := d r=1 (−1) r−1 /r!. The result holds without any restriction on the characteristic of Fq and provides an explicit expression for the constant underlying the O-notation in terms of d. We reduce the question to estimating the number of Fq-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over Fq. For this purpose, we obtain an upper bound on the dimension of the singular locus of the complete intersections under consideration, which allows us to estimate the corresponding number of Fq-rational points.

Finite Fields and Their Applications, 2004

We generalize a recent idea for constructing codes over a finite field F q by evaluating a certain collection of polynomials over F q at elements of an extension field. We show that many codes with the best parameters presently known can be obtained by this construction. In particular, a new linear code, a ½40; 23; 10-code over F 5 is discovered. Moreover, several families of optimal and near-optimal codes can also be obtained by this method. We call a code near-optimal if its minimum distance is within 1 of the known upper bound.

Journal of Combinatorial Theory, Series A, 2014

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F q [T ] of degree d for which s consecutive coefficients a d−1 ,. .. , a d−s are fixed. Our estimate holds without restrictions on the characteristic of F q and asserts that V(d, s, a) = µ d q + O(1), where V(d, s, a) is such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 ,. .. , d d−s). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of F q-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of F q-rational points is established.

Discrete Mathematics & Theoretical Computer Science

We present two results, the first on the distribution of the roots of a polynomial over the ring of integers modulo n and the second on the distribution of the roots of the Sylvester resultant of two multivariate polynomials. The second result has application to polynomial GCD computation and solving polynomial diophantine equations.

Designs, Codes and Cryptography, 2008

In this paper, we study the p-ary linear code C(PG(n,q)), q = p h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound (Geom Dedicata 8:407–415, 1979).

arXiv (Cornell University), 2013

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F q [T ] of degree d for which s consecutive coefficients a d-1 , . . . , a d-s are fixed. Our estimate asserts that V(d, s, a) = µ d q + O(q 1/2 ), where V(d, s, a) is such an average cardinality, µ d := d r=1 (-1) r-1 /r! and a := (a d-1 , . . . , a d-s ). We also prove that V 2 (d, s, a) = µ 2 d q 2 +O(q 3/2 ), where that V 2 (d, s, a) is the average second moment on any family of monic polynomials of F q [T ] of degree d with s consecutive coefficients fixed as above. Finally, we show that V 2 (d, 0) = µ 2 d q 2 + O(q), where V 2 (d, 0) denotes the average second moment of all monic polynomials in F q [T ] of degree d with f (0) = 0. All our estimates hold for fields of characteristic p > 2 and provide explicit upper bounds for the constants underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the questions to estimate the number of F q -rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q . A critical point for our results is an analysis of the singular locus of the varieties under consideration, which allows to obtain rather precise estimates on the corresponding number of F q -rational points.

2017

Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well known. We consider a class $\mathcal{F}_{q,n}$ of polynomials, which we obtain by modifying linear permutations at $n$ points. The study of the spectrum of $\mathcal{F}_{q,n}$ enables us to obtain a simple description of polynomials $F \in \mathcal{F}_{q,n}$ with prescribed $V_F$, especially those avoiding a given set, like cosets of subgroups of the multiplicative group $\mathbb{F}_q^*$. The value set count for such $F$ can also be determined. This yields polynomials with evenly distributed values, which have small maximum count.

Discrete Mathematics, 2022

Linear sets on the projective line have attracted a lot of attention because of their link with blocking sets, KM-arcs and rank-metric codes. In this paper, we study linear sets having two points of complementary weight, that is with two points for which the sum of their weights equals the rank of the linear set. As a special case, we study those linear sets having exactly two points of weight greater than one, by showing new examples and studying their equivalence issue. Also we determine some linearized polynomials defining the linear sets recently introduced by Jena and Van de Voorde (2021).

Combinatorica, 2016

We obtain estimates on the number |A λ | of elements on a linear family A of monic polynomials of Fq[T ] of degree n having factorization pattern λ := 1 λ 1 2 λ 2 • • • n λn. We show that |A λ | = T (λ) q n−m +O(q n−m−1/2), where T (λ) is the proportion of elements of the symmetric group of n elements with cycle pattern λ and m is the codimension of A. Furthermore, if the family A under consideration is "sparse", then |A λ | = T (λ) q n−m + O(q n−m−1). Our estimates hold for fields Fq of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the O-notation in terms of λ and A with "good" behavior. Our approach reduces the question to estimate the number of Fq-rational points of certain families of complete intersections defined over Fq. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of Fq-rational points are established.