Linear cyclic codes of wordlength v over GF(q s ) which are also linear cyclic codes of wordlength sv over GF(q)

Any nonsingular linear transformation : GF(q^s) GF(q^s) can be used to treat a linear cyclic code of wordlength v over GF(q^s) as a linear code () of Wordlength sv over GF(q). This paper determines those linear cyclic codes and transformations for which the resulting linear code () is also cyclic.     

Existence of (q, 7, 1) difference families with q a prime power

The existence of a (q,k, 1) difference family in GF(q) has been completely solved for k = 3,4,5,6. For k = 7 only partial results have been given. In this article, we continue the investigation and use Weil's theorem on character sums to show that the necessary condition for the existence of a (q,7,1) difference family in GF(q), i.e. q ≡ 1; (mod 42) is also sufficient except for q = 43 and possibly except for q = 127, q = 211, q = 31⁶ and primes q∈ [261239791, 1.236597 × 10¹³] such that in GF(q). © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 126–138, 2002; DOI 10.1002/jcd.998     

Embeddings of Un(q2) and symmetric strongly regular graphs

In the geometric setting of the embedding of the unitary group U_n(q²) inside an orthogonal or a symplectic group over the subfield GF(q) of GF(q²), q odd, we show the existence of infinite families of transitive two‐character sets with respect to hyperplanes that in turn define new symmetric strongly regular graphs and two‐weight codes. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 248–253, 2010     

The explicit construction of irreducible polynomials over finite fields

For a finite field GF(q) of odd prime power order q, and n 1, we construct explicitly a sequence of monic irreducible reciprocal polynomials of degree n2 ^m (m = 1, 2, 3, ...) over GF(q). It is the analog for fields of odd order of constructions of Wiedemann and of Meyn over GF(2). We also deduce iterated presentations of GF (q ⁿ 2).     

A New Family of 2-Designs over GF (q) Admitting SLm(q1)

Given a 2-(l,3,q³(q^l-5-1/q-1);q) design for an integer l 5 mod 6(q-1) which admits the action of a Singer cycle Z_l of GL_l(q), we construct a 2-(ml,3,q³(q^l-5-1/q-1);q) design for an arbitrary integer m 3 which admits the action of SL_m(q^l). The construction applied to Suzuki's designs actually provides a new family of 2-designs over GF(q) which admit the SL_m(q^l) action.     

A generalized Gleason–Pierce–Ward theorem

The Gleason–Pierce–Ward theorem gives constraints on the divisor and field size of a linear divisible code over a finite field whose dimension is half of the code length. This result is a departure point for the study of self-dual codes. In recent years, additive codes have been studied intensively because of their use in additive quantum codes. In this work, we generalize the Gleason–Pierce–Ward theorem on linear codes over GF(q), q = p ^m , to additive codes over GF(q). The first step of our proof is an application of a generalized upper bound on the dimension of a divisible code determined by its weight spectrum. The bound is proved by Ward for linear codes over GF(q), and is generalized by Liu to any code as long as the MacWilliams identities are satisfied. The trace map and an analogous homomorphism on GF(q) are used to complete our proof.      

On translation planes of orderq 3 that admit a collineation group of orderq 3, II

Let II be a translation plane of orderq ³ with kernel GF(q) that admits a collineation groupG of orderq ³ in the linear translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity.In this paper, we show that any such translation plane II is one of the following types of planes:     

On the size of the Jacobians of curves over finite fields

Abdract  Given a smooth curve of genus g ≥ 1 which admits a smooth projective embedding of dimension m over the ground field of q elements, we obtain the asymptotic formula q ^g+o(g) for the size of set of the -rational points on its Jacobian in the case when m and q are bounded and g → ∞. We also obtain a similar result for curves of bounded gonality. For example, this applies to the Jacobian of a hyperelliptic curve of genus g → ∞.      

Reed-Muller codes and Barnes-Wall lattices: Generalized multilevel constructions and representation over GF(2<Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>)

Generalized multilevel constructions for binary RM(r,m) codes using projections onto GF(2 ^q ) are presented. These constructions exploit component codes over GF(2), GF(4),..., GF(2 ^q ) that are based on shorter Reed-Muller codes and set partitioning using partition chains of length-2 ^l codes. Using these constructions we derive multilevel constructions for the Barnes-Wall Λ(r,m) family of lattices which also use component codes over GF(2), GF(4),..., GF(2 ^q ) and set partitioning based on partition chains of length-2 ^l lattices. These constructions of Reed-Muller codes and Barnes-Wall lattices are readily applicable for their efficient decoding.      

On an Isomorphism Problem of Some Dual Hyperovals in PG(2d + 1, q) with q even

Let q = 2^l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qⁿ) with n≥ d + 1 and for any generator σ of the Galois group of GF(qⁿ) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(q^d+1), two dual hyperovals and in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(q^d+1) over GF(q), are isomorphic if and only if (1) σ = τ or (2) σ τ = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3.     

Some (17q, 17, 2) and (25q, 25, 3) BIBD Constructions

We give the explicit construction of a regular (17q, 17, 2)-BIBD for any prime power q 17 (mod 32) such that 2 is not a 4th power in GF(q) and the explicit construction of a regular (25q, 25, 3)-BIBD for any prime power q 25 (mod 48) such that and +3 are non-squares in GF(q).     

A nowhere-zero point in linear mappings

We state the following conjecture and prove it for the case whereq is a proper prime power:Let A be a nonsingular n by n matrix over the finite field GF_qq4, then there exists a vector x in (GF_q)ⁿ such that both x and Ax have no zero component.Research supported in part by Allon Fellowship and by a Bat Sheva de Rothschild grant.     

On a Class of Symmetric Balanced Generalized Weighing Matrices

Let q be a prime power and m a positive integer. A construction method is given to multiply the parametrs of an -circulant BGW(v=1+q+q ₂+·+q ^m , q ^m , q ^m –q ^m–1) over the cyclic group C _n of order n with (q–1)/n being an even integer, by the parameters of a symmetric BGW(1+q ^m+1, q ^m+1, q ^m+1–q ^m ) with zero diagonal over a cyclic group C _vn to generate a symmetric BGW(1+q+·+q ^2m+1,q ^2m+1,q ^2m+1–q ^2m) with zero diagonal, over the cyclic group C _n . Applications include two new infinite classes of strongly regular graphs with parametersSRG(36(1+25+·+25^2m+1),15(25)^2m+1,6(25)^2m+1,6(25)^2m+1), and SRG(36(1+49+·+49^2m+1),21(49)^2m+1,12(49)^2m+1,12(49)^2m+1).     

Recurrent Methods for Constructing Irreducible Polynomials over GF(2s)

The paper is devoted to some results concerning the constructive theory of the synthesis of irreducible polynomials over Galois fields GF(q), q=2^s. 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.     

Combinatorial geometries representable over GF(3) and GF(q). II. Dowling geometries

Letq be an odd prime power not divisible by 3. In Part I of this series, it was shown that the number of points in a rank-n combinatorial geometry (or simple matroid) representable over GF(3) and GF(q) is at mostn ². In this paper, we show that, with the exception ofn = 3, a rank-n geometry that is representable over GF(3) and GF(q) and contains exactlyn ² points is isomorphic to the rank-n Dowling geometry based on the multiplicative group of GF(3).This research was partially supported by the National Science Foundation under Grants DMS-8521826 and DMS-8500494.     

Packing problems in Galois geometries over GF(3)

A set of kind s in the Galois space S _r,q is a set of points such that any s+1 are linearly independent but there is at least one subset of s+2 The packing problem is that of finding , the largest size of kind s in S _r,q. The main result is the evaluation of for all sr5. linearly dependent points. Some partial results bounding m ^s _6,3 are also given.     

Cyclotomic numbers and primitive idempotents in the ring GF(q)[x]/(x−1)

Let q be an odd prime power and p be an odd prime with gcd(p,q)=1. Let order of q modulo p be f, and q^f=1+pλ. Here expressions for all the primitive idempotents in the ring R_pⁿ=GF(q)[x]/(x^pn−1), for any positive integer n, are obtained in terms of cyclotomic numbers, provided p does not divide λ if n2. The dimension, generating polynomials and minimum distances of minimal cyclic codes of length pⁿ over GF(q) are also discussed.     

Combinatorial geometries representable over GF(3) and GF(q). I. The number of points

Letq be a prime power not divisible by 3. We show that the number of points (or rank-1 flats) in a combinatorial geometry (or simple matroid) of rankn representable over GF(3) and GF(q) is at mostn ². Whenq is odd, this bound is sharp and is attained by the Dowling geometries over the cyclic group of order 2.This research was partially supported by National Science Foundation Grant DMS-8521826 and a North Texas State University Faculty Research Grant.     

The 2‐Blocking Number and the Upper Chromatic Number of PG(2,q)

A twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ₂(Π). Let PG(2,q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF(q), then τ₂PG(2,q))≤ 2(q+(q‐1)/(r‐1)). For a finite projective plane Π, let denote the maximum number of classes in a partition of the point‐set, such that each line has at least two points in some partition class. It can easily be seen that (?) for every plane Π on v points. Let , p prime. We prove that for , equality holds in (?) if q and p are large enough.