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.     

The weight distribution of linear codes over GF(ql) having generator matrix over GF(q>)

We study the weight distribution of the linear codes over GF(q^l) which have generator matrices over GF(q) and their dual codes. As an application we find the weight distribution of the irreducible cyclic (23(2¹≈1), 111) codes over GF(2) for all lnot divisible by 11.     

Factorization of the Cyclotomic Polynomialx+ 1 over Finite Fields

The aim of this note is to show that the (well-known) factorization of the 2ⁿ⁺¹th cyclotomic polynomialx2ⁿ+ 1 over GF(q) withq≡ 1 (mod 4) can be used to prove the (more complicated) factorization of this polynomial over GF(q) withq≡ 3 (mod 4).     

New linear codes over GF(8)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d₈(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 8]-code for given values of n and k. In this paper, eighteen new linear codes over GF(8) are constructed which improve the table of d₈(n, k) by Brouwer.     

Irreducible polynomials over GF(2) with prescribed coefficients

For an even positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF(2) in which the coefficients of xⁿ⁻¹,xⁿ⁻² and xⁿ⁻³ are specified in advance. Formulas for the number of elements in GF(2ⁿ) with the first three traces specified are also given.     

2-Designs overGF(q)

We give a construction of a series of 2-(n, 3,q ²+q+1;q) designs of vector spaces over a finite fieldGF(q) of odd characteristic. These designs correspond to those constructed by Thomas and the author for even characteristic. As a natural generalization we give a collection ofm-dimensional subspaces which possibly become a 2-(n, m, λ; q) design.     

Irreducible polynomials over GF(2) with three prescribed coefficients

For an odd positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF(2) in which the coefficients of xⁿ⁻¹, xⁿ⁻² and xⁿ⁻³ are specified in advance. Formulas for the number of elements in GF(2ⁿ) with the first three traces specified are also given.     

A conversion algorithm for logarithms on GF(2n)

We present a method for expressing a root of one irreducible polynomial of degree n over GF(2) in terms of a basis of GF(2ⁿ) over GF(2) associated with another. This allows us, when both polynomials are primitive, to find logarithms relative to one polynomial from logarithms relative to the other.     

Some results concerning cryptographically significant mappings over GF(2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper we investigate the existence of permutation polynomials of the form F(x) = x ^d  + L(x) over GF(2 ⁿ ), L being a linear polynomial. The results we derive have a certain impact on the long-term open problem on the nonexistence of APN permutations over GF(2 ⁿ ), when n is even. It is shown that certain choices of exponent d cannot yield APN permutations for even n. When n is odd, an infinite class of APN permutations may be derived from Gold mapping x ³ in a recursive manner, that is starting with a specific APN permutation on GF(2 ^k ), k odd, APN permutations are derived over GF(2 ^k+2i ) for any i ≥ 1. But it is demonstrated that these classes of functions are simply affine permutations of the inverse coset of the Gold mapping x ³. This essentially excludes the possibility of deriving new EA-inequivalent classes of APN functions by applying the method of Berveglieri et al. (approach proposed at Asiacrypt 2004, see [3]) to arbitrary APN functions.     

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.     

Geometry of Cubic and Quartic Hypersurfaces over Finite Fields

Let YPⁿ be a cubic hypersurface defined over GF(q). Here, we study the Finite Field Nullstellensatz of order [q/3] for the set Y(q) of its GF(q)-points, the existence of linear subspaces of PG(n,q) contained in Y(q) and the possibility to join any two points of Y(q) by the union of two lines of PG(n,q) entirely contained in Y(q). We also study the existence of linear subspaces defined over GF(q) for the intersection of Y with s quadrics and for quartic hypersurfaces.     

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).     

Optimal pseudo-cyclic codes and caps in PG (3,q)

Existence and uniqueness of pseudo-cyclic [q ²+1,q ²–3, 4]-codes over GF(q) are proved. Elliptic quadrics are characterized as those (q ²+1)-caps in PG(3,q) whose corresponding [q ²+1,q ²–3, 4]-codes are pseudo-cyclic.     

The number of equivalence classes of nondegenerate bilinear and sesquilinear forms over a finite field

Generating functions in the form of infinite products are given for the number of equivalence classes of nondegenerate sesquilinear forms of rank n over GF(q²) and for the number of equivalence (or congruence) classes of nondegenerate bilinear forms of rank n over GF(q).     

Directed graph representation of half-rate additive codes over GF(4)

We show that (n, 2 ⁿ ) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n, 2 ⁿ ) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.     

Flag transitive planes of orderq n with a long cyclel ∞ as a collineation

Baker and Ebert [1] presented a method for constructing all flag transitive affine planes of orderq ² havingGF(q) in their kernels for any odd prime powerq. Kantor [6; 7; 8] constructed many classes of nondesarguesian flag transitive affine planes of even order, each admitting a collineation, transitively permuting the points at infinity. In this paper, two classes of non-desarguesian flag transitive affine planes of odd order are constructed. One is a class of planes of orderq ⁿ , whereq is an odd prime power andn 3 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. The other is a class of planes of orderq ⁿ , whereq is an odd prime power andn 2 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. Since each plane of the former class is of odd dimension over its kernel, it is not isomorphic to any plane constructed by Baker and Ebert [1]. The former class contains a flag transitive affine plane of order 27 constructed by Kuppuswamy Rao and Narayana Rao [9]. Any plane of the latter class of orderq ⁿ such thatn 1 (mod 2), is not isomorphic to any plane constructed by Baker ad Ebert [1].The author is grateful to the referee for many helpful comments.     

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.     

On the sharpness of a theorem of B. segre

The theorem of B. Segre mentioned in the title states that a complete arc of PG(2,q),q even which is not a hyperoval consists of at mostq−√q+1 points. In the first part of our paper we prove this theorem to be sharp forq=s ² by constructing completeq−√q+1-arcs. Our construction is based on the cyclic partition of PG(2,q) into disjoint Baer-subplanes. (See Bruck [1]). In his paper [5] Kestenband constructed a class of (q−√q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband’s and our constructions. We prove that these constructions result in isomorphic (q−√q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF(q ³) over GF(q), and on a representation of GF(q)³ in GF(q ³)³ indicated in Jamison [4].     

Some families of complete caps of quadrics and symplectic polarities over GF(8)

A cap on a quadric is a set of its points whose pairwise joins are all chords. A cap is complete if it is not part of a larger one. The only field for which all complete quadric caps are known is GF(2). Those caps are small; the biggest for each quadric is of order the dimension of the ambient space. Apart from information about ovoids in dimensions at most 7, little else is known. Here, the evidence is increased by providing caps over GF(2), odd, which, if >1, have size of order the dimension cubed. In particular, complete caps are obtained for the quadrics Q _2m (8), Q ₊ ^8k+7 (8), Q _- ^8k+3 (8), Q ₊ ^8k+1 (8) and Q _- ^8k+5 (8). These caps on Q ₊ ^8k+7 (8) and Q _- ^8k+3 (8) are complete on any Q _n(8) of which their quadrics are sections; so is that that of Q ₄₊₂(8) for any Q _2n (8) of which Q ₄₊₂(8) is a section with the same kernel. From the correspondence with Q _2n (8) complete caps are obtained for symplectic polarities over GF(8).     

ExplicitN-polynomials of 2-power degree over finite fields,I

For an odd prime powerq the infinite field GF(q ² )= _n0 GF (q ²ⁿ ) is explicitly presented by a sequence (f _n)₁ ofN-polynomials. This means that, for a suitably chosen initial polynomialf ₁, the defining polynomialsf _nGF(q)[x] of degrees2 ⁿ are constructed by iteration of the transformation of variablexx+1/x and have linearly independent roots over GF(q). In addition, the sequences are trace-compatible in the sense that the relative traces map the corresponding roots onto each other. In this first paper the caseq1 (mod 4) is considered and the caseq3 (mod 4) will be dealt with in a second paper. This specific construction solves a problem raised by A. Scheerhorn in [11].