Finite Fields and Their Applications

In this paper we construct some algebraic geometric error-correcting codes on surfaces whose Neron-Severi group has low rank. If the Neron-Severi group is generated by an effective divisor, the intersection of this surface with an irreducible surface of ...

We study subsets of Grassmann varieties G(l,m) over a field F, such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study unions of Schubert cycles of Grassmann varieties G(l,m) over a field F. We compute their linear ...

We describe the Galois closure of the Garcia-Stichtenoth tower and prove that it is optimal.

We present a new construction for (n,w,@l)-optical orthogonal codes (OOCs). The construction is pleasingly simple, where codewords correspond to arcs, specifically normal rational curves. Moreover, our construction yields for each @l>1 an infinite ...

In this note, we study relative (p^a,p^b,p^a,p^a^-^b)-relative difference sets in certain p-subgroups of SL(n,K), K=F"q, where q is a prime power.

In 1988 Garcia and Voloch proved the upper bound 4n^4^/^3(p-1)^2^/^3 for the number of solutions over a prime finite field F"p of the Fermat equation x^n+y^n=a, where a@?F"p^* and n>=2 is a divisor of p-1 such that (n-12)^4>=p-1. This is better than ...

Let K be a finite extension of F"2. We consider quadratic forms written as the trace of xR(x), where R(x) is a linearized polynomial. We determine the K and R(x) where the form has a radical of codimension 2. This is applied to constructing maximal ...

We show that, if p<>3 is an odd prime satisfying p@__ __5(mod8), then each nonzero element of GF(p) can be written as a sum of distinct quadratic residues in the same number of ways, N say, and that the number of ways of writing 0 as a sum of distinct ...

Miyaji, Nakabayashi and Takano (MNT) gave families of group orders of ordinary elliptic curves with embedding degree suitable for pairing applications. In this paper we generalise their results by giving families corresponding to non-prime group orders. ...

We construct a family of simple 3-(2^m,8,14(2^m-8)/3) designs, with odd m>=5, from all Z"4-Goethals-like codes G"k. In addition, these designs imply the existence of other design families with the same parameters as the designs constructed from the Z"4-...

Let e be a positive integer, p be an odd prime, q=p^e, and F"q be the finite field of q elements. Let f"2,f"3@__ __F"q[x,y]. The graph G=G"q(f"2,f"3) is a bipartite graph with vertex partitions P=F"q^3 and L=F"q^3, and edges defined as follows: a vertex ...

Let 0<@a"1<...<@a"k be integers and f(x)=@__ __"i"="1^ka"ix^2^^^@a^^^"^^^i^+^1+bx@__ __F"2"^"m[x], a"k<>0. Define S(f,n)=@__ __"x"@__ __"F"""2"""^"""ne(f(x)) where m|n and e(x)=(-1)^T^r^"^F^"^"^"^2^"^"^"^^^"^"^"^n^"^/^"^F^"^"^"^2^(^x^). We establish a ...

Permutation polynomials have been a subject of study for over 140 years and have applications in many areas of science and engineering. However, only a handful of specific classes of permutation polynomials are known so far. In this paper we describe ...

Guruswami-Sudan algorithm for polynomial reconstruction problem plays an important role in the study of error-correcting codes. In this paper, we study new better parameter choices in Guruswami-Sudan algorithm for the polynomial reconstruction problem. ...

Let N be the number of solutions of the equationx"1^m^"^1+...+x"n^m^"^n=ax"1...x"n over the finite field F"q=F"p"^"s. L. Carlitz found formulas for N when n=3 or 4, m"1=...=m"n=2. In an earlier paper, we obtained formulas for N when d=gcd(@?"j"="1^nm/m"...

Let X->Y^0 be an abelian prime-to-p Galois covering of smooth schemes over a perfect field k of characteristic p>0. Let Y be a smooth compactification of Y^0 such that Y-Y^0 is a normal crossings divisor on Y. We describe a logarithmic F-crystal on Y ...

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and ...

We obtain sharp estimates for p-adic oscillatory integrals of the formE"A(z,f)=@!A@j(@__ __j=1lz"jf"j(x))|dx|, where @j is a nontrivial additive character on a non-archimedean local field K of arbitrary characteristic, and f=(f"1,...,f"l):A->K^l is a ...

We introduce an invariant for nonsingular quadratic forms that take values in a Galois Ring of characteristic 4. This notion extends the invariant in Z"8 for Z"4-valued quadratic forms defined by Brown [E.H. Brown, Generalizations of the Kervaire ...

From a rational convex polytope of dimension r>=2 J.P. Hansen constructed an error correcting code of length n=(q-1)^r over the finite field F"q. A rational convex polytope is the same datum as a normal toric variety and a Cartier divisor. The code is ...

We describe some relations on the coefficients of a polynomial in terms of the map that induces and use them to characterize the coefficients of the inverse polynomials of some special classes of permutation polynomials.

It is easy to see that an element P(t)@__ __F"2[t] is a sum of cubes if and only ifP(t)@__ __M(2):={P(t):P(t)=0 or 1(modt^2+t+1)}. We say that P(t) is a ''strict'' sum of cubes A"1(t)^3+...+A"g(t)^3 if we have deg(A"i^3)=

Complete n-tracks in PG(N,q) and non-extendable Near MDS codes of dimension N+1 over F"q are known to be equivalent objects. The best known lower bound for the maximum number of points of an n-track is attained by elliptic n-tracks, that is, n-tracks ...

Let p and 2p+3 be prime powers and p=3(mod 4). We describe a construction of a symmetric design D with parameters (4(p+1)^2,2p^2+3p+1,p^2+p). If p and 2p+3 are primes, then a derived design of D is 1-rotational.

I present some results towards a complete classification of monomials that are Almost Perfect Nonlinear (APN), or equivalently differentially 2-uniform, over F"2"^"n for infinitely many positive integers n. APN functions are useful in constructing S-...

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 @?n2@? coefficients prescribed, where the first ...

In this paper we study construction algorithms for polynomial lattice rules modulo arbitrary polynomials. Polynomial lattice rules are a special class of digital nets which yield well distributed point sets in the unit cube for numerical integration. ...

Polyadic codes constitute a special class of cyclic codes and are generalizations of quadratic residue codes, duadic codes, triadic codes, m-adic residue codes and split group codes, which have good error-correcting properties. In this paper, we give ...

Let m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2^m over F"q are determined explicitly.