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.     

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.     

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.     

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.     

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.     

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.      

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.     

The Weight Enumerator of Linear Codes over GF q m) Having Generator Matrix over GF q

We have the relationships between the Hamming weight enumerator of linear codes over GFq ^m which have generator matrices over GFq, the support weight enumerator and the -ply weight enumerator.     

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.     

Symmetry codes over GF(3)

A lower bound $\text{{}\text{1}\text{4}\text{(4q + 5)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+}\text{3}\text{2}$ is given for the minimum weight of the symmetry code C(q) over GF(3), which is introduced by Pless [3].     

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.     

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.     

The weight distribution of irreducible cyclic codes with block lengths n1((q1−1)N)

We study the weight distribution of irreducible cyclic (n, k) codeswith block lengths n = n₁((q¹ ? 1)/N), where N|q ? 1, gcd(n₁,N) = 1, and gcd(l,N) = 1. We present the weight enumerator polynomial, A(z), when k = n₁l, k = (n₁ ? 1)l, and k = 2l. We also show how to find A(z) in general by studying the generator matrix of an (n₁, m) linear code, $\text{V}{}^1{}_{\mathrm{d}}$ over GF(q^d) where d = gcd (ord_n1(q), l). Specifically we study A(z) when $\text{V}{}^1{}_{\mathrm{d}}$ is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers A_μ which completely determine the weight distributionof any irreducible cyclic (n₁(2¹ ? 1), k) code over GF(2) for all n₁ ? 17.     

The existence of maximal (q 2, 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic

We prove that (q ², 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q ² and prime characteristic. For odd prime characteristic, our construction solves the maximal arc problem. For characteristic 2, an extension of the above construction yields the lower bound q ² + 2 on the maximum size of a 2-arc in PHG(2, R). Translating the arcs into codes, we get linear [q ³, 6, q ³ ?q ² ?q] codes over ${\mathbb {F}_q}$ for every prime power q > 1 and linear [q ³ + q, 6,q ³ ?q ² ?1] codes over ${\mathbb {F}_q}$ for the special case q = 2 ^r . Furthermore, we construct 2-arcs of size (q + 1)²/4 in the planes PHG(2, R) over Galois rings R of length 2 and odd characteristic p ².     

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).     

Orbits under actions of affine groups over GF (2)

Let G be a group (or vector space) and A a group of transformations of G. A then acts as a group of transformations of P(G), the set of subsets of G. It is meaningful to study the orbit structure of P(G) under the action of A. The question of the existence of elements of P(G) with trivial isotropy subgroup seems to be of interest in studying the action of A on G. In this paper actions of affine groups over GF (2) are considered. It is proved, by an inductive construction, that every vector space over GF (2) of dimension at least six contains a subset with trivial isotropy subgroup.     

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).     

On subgroups and factor groups of GL(n, q) acting on spreads with the same characteristic

Let π^l_∞ be an affine translation plane of order q^r with GF(q) in its kern. Suppose G is a subgroup of the translation complement of π^l_∞ which leaves invariant a set Δ of q + 1 slopes and acts transitively on l_∞?Δ. We study the situation when G≌SL(n, q) or PSL(n, q).We show that if G|Δ = identity, then π^l_∞ is a Hall plane, a Lorimer-Rahilly plane (LR-16) or a Johnson-Walker plane (JW-16). Moreover, if n?3, then G fixes Δ elementwise and π^l_∞ is LR-16 or JW-16.     

Vectorial Resilient PC（l） of Order k Boolean Functions from AG-Codes

Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose (see [1–4, 7, 8, 10, 11, 16]). Kurosawa, Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying PC(l) of order k from binary linear or nonlinear codes. In this paper, the algebraic-geometric codes over GF(2m) are used to modify the Carlet and Kurosawa-Satoh’s construction for giving vectorial resilient Boolean functions satisfying PC(l) of order k criterion. This new construction is compared with previously known results.     

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.