Matroid representation over GF(3)

We show that a matroid is representable over GF(3) if and only if no minor is the five-point line or the Fano matroid, or their duals. Tutte's famous characterization of the regular matroids is a corollary. A key lemma states that two representations of the same matroid in the same vector space over GF(3) may be transformed one into the other by inverting some points through the origin and taking a linear transformation; no result of this kind holds in larger fields.     

On critical exponents of Dowling matroids

The critical problem in matroid theory is the problem to determine the critical exponent of a given representable matroid over a finite field. In this paper, we study the critical exponents of a class of representable matroids over finite fields, called Dowling matroids. Then the critical problem for a Dowling matroid is corresponding to the classical problem in coding theory to determine the maximum dimension k such that there exists an \([n,k,d]_q\) code for given n, d and q. We give a necessary and sufficient condition on the critical exponents of Dowling matroids by using a coding theoretical approach.     

Author index for volume 25

We extend to all regular matroids the fact that the chromatic number of a graph is k or less, when all vertex degrees are less than k. If there is a covering of a regular matroid by cocircuits of cardinality less than k, then the chromatic number of the matroid is k or less. A similar result holds for the critical exponent of a vector matroid over GF(q).     

Vector representable matroids of given rank with given automorphism group

A simple way of associating a matroid of prescribed rank with a graph is shown. The matroids so constructed are representable over any sufficiency large field. Their use is demonstrated by the following result: Given an integer k?3 and a function G associating a group with each subset of a set S, there is a matroid M(E), representable over any sufficiently large field, such that E ? S, and for any T ?/ S, the rank of M/Tis k, and the automorphine group of MT is isomorphic to G(T).     

Inequivalent representations of matroids over prime fields

It is proved that for each prime field GF(p)$\mathit{GF}(p)$, there is an integer _np$n_p$ such that a 4-connected matroid has at most _np$n_p$ inequivalent representations over GF(p)$\mathit{GF}(p)$. We also prove a stronger theorem that obtains the same conclusion for matroids satisfying a connectivity condition, intermediate between 3-connectivity and 4-connectivity that we term “k-coherence”.     

Non-regular simplicial matroids

Let (ks) denote the set of all k-element-subsets of a finite set S. A k-simplical matroid on a subset E of (ks) is a binary matroid the circuit of which are simplicial complexes {X1,…Xm} ? E with boundary 0 (mod 2). The k-simplical matroid on (ks) is called the full simplicial matroid Gk(S). The polygon matroid on the edges of a finite graph is 2-simplicial. Polygon-matroids and their duals are regular. The dual of Gk(S) is Gn?k(S) if the cardinnlity of S is n. More details on simplicial matroids can be found in [3, Chapter 6] and also in [4, pp. 180–181].Welsh asked if every simplicial matroid is regular. We prove that this is not the case, for all full k-simplicial matroids Gk(S) with 3?k?n?3 are non-regular (n is the cardinality of S). This result has also been proved σy R. Cordovil and M. Las Vergnas recently. Their proof is different from our proof, which is somewhat shorter.     

Periodicities of partition functions and stirling numbers modulo p

If p(n, k) is the number of partitions of n into parts ≤k, then the sequence {p(k, k), p(k + 1, k),…} is periodic modulo a prime p. We find the minimum period Q = Q(k, p) of this sequence. More generally, we find the minimum period, modulo p, of {p(n; T)}_{n ≥ 0}, the number of partitions of n whose parts all lie in a fixed finite set T of positive integers. We find the minimum period, modulo p, of {S(k, k), S(k + 1, k),…}, where these are the Stirling numbers of the second kind. Some related congruences are proved. The methods involve the use of cyclotomic polynomials over Z_p[x].     

Finite prime-field characteristic sets for planar configurations

Examples are constructed of planar matroids with finite prime-field characteristic sets (i.e. matroids representable over a finite set of prime fields but over fields of no other characteristic). In particular, for any n>3, a projectively unique integer matrix is constructed with 2lsqblog₂nrsqb+6 columns which often gives nonsingleton characteristic sets and, when n is prime, has characteristic set {n}. Many finite subsets of primes are shown to be characteristic sets, including {23,59} (the smallest pair found using these methods), all pairs of primes {p, p′:67?p<p′?293}, and the seventeen largest five-digit primes. Probabilistic arguments are presented to support the conjecture that prime-field characteristic sets exist of every finite cardinality. For p>3, AG(2,p) is shown to be a subset of PG(2, q) only for q=p^s. Another general construction technique suggests that when $\text{P}\text{={p}{}_1\text{,…,p}{}_{\mathrm{k}}\text{}}$ are the primitive prime divisors of 2ⁿ±1 (n sufficiently large), then there is a matroid with O (log n) points whose characteristic set is $\text{P}$. We remark that although only one finite nonsingleton characteristic set (due to R. Reid) was known prior to this paper, a new technique by J. Kahn has shown that every finite set of primes forms a (non-prime-field) characteristic set.     

Self-Orthogonal and Self-Dual Codes Constructed via Combinatorial Designs and Diophantine Equations

Combinatorial designs have been widely used, in the construction of self-dual codes. Recently, new methods of constructing self-dual codes are established using orthogonal designs (ODs), generalized orthogonal designs (GODs), a set of four sequences and Diophantine equations over GF(p). These methods had led to the construction of many new self-dual codes over small finite fields and rings. In this paper, we used some methods to construct self-orthogonal and self dual codes over GF(p), for some primes p. The construction is achieved by using some special kinds of combinatorial designs like orthogonal designs and GODs. Moreover, we combine eight circulant matrices, a system of Diophantine equations over GF(p), and a recently discovered array to obtain a new construction method. Using this method new self-dual and self-orthogonal codes are obtained. Specifically, we obtain new self-dual codes [32,16,12] over GF(11) and GF(13) which improve the previously known distances.     

Representations of matroids and free resolutions for multigraded modules

Let k be a field, let R=k[x₁,…,xm] be a polynomial ring with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, and let be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T_•(Φ,S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation ? over k of a matroid M a canonical finite complex of finite dimensional k-vector spaces T_•(?) that is a resolution of Ker?. We also show that the length of T_•(?) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map ?.     

Factorization of Q(h(T)(x)) over a finite field where Q(x) is irreducible and h(T)(x) is linear—I

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(q)(x), and let h(T)(x) be a linear polynomial in GF(q)[x], where T:x→x^q. We use properties of the linear operator h(T) to give conditions for Q(h(T)(x)) to have a root of arbitrary degree k over GF(q), and we describe how to count the irreducible factors of Q(h(T)(x)) of degree k over GF(q). In addition we compare our results with those Ore which count the number of irreducible factors belonging to a linear polynomial having index k.     

A New Characterization of Semi-bent and Bent Functions on Finite Fields*

We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF(2)-linear combination of Gold functions Tr(x²ⁱ+1) is semi-bent over GF(2ⁿ), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold functions give rise to semi-bent functions over GF(2^p) when p belongs to a certain class of primes. Second, we generalize our results to fields GF(pⁿ) where p is an odd prime and n is odd. In that case, we can determine whether a GF(p)-linear combination of Gold functions Tr(x^pi+1) is (generalized) semi-bent or bent by a polynomial GCD computation. Similar to the binary case, simple characterizations of these p-ary semi-bent and bent functions are provided. Parts of this paper were presented at the 2002 IEEE International Symposium on Information Theory [10]     

Non-cover generalized Mycielski, Kneser, and Schrijver graphs

A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. We prove that the generalized Mycielski graphs M_m(C_2t+1) of an odd cycle, Kneser graphs KG(n,k), and Schrijver graphs SG(n,k) are not cover graphs when m?0,t?1, k?1, and n?2k+2. These results have consequences in circular chromatic number.     

Enumeration of matrices of given rank with submatrices of given rank

The authors determine the number of (n+m)×t matrices A¹ of rank r+v, over a finite field GF(q), whose last m rows are those of a given matrix A of rank r+v over GF(q) and whose first n rows have a given rank u.     

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 Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field

We prove that for every field k and every positive integer n there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k, n) denote the fraction of the isogeny classes of n-dimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every n we have S(F_q, n)→1 as q→∞ over the prime powers.     

Diagonal forms over finite fields

We shall establish for all finite fields GF(pⁿ) the following result of Chowla: given a positive integer m greater than one and the finite field GF(p), p a prime, such that x^m = ?1 is solvable in GF(p), then there exists an absolute positive constant c, $\text{c ≤}\text{10}\text{ln}\text{2}$, such that for each set of s nonzero elements a_i of GF(p), $\text{a}{}_1\text{x}{}_1{}^{\mathrm{m}}\text{+ ? + a}{}_{\mathrm{s}}\text{x}{}_{\mathrm{s}}{}^{\mathrm{m}}$ has a non-trivial zero in GF(p) if s ≥ c ln m.     

Uniform cyclotomy

Let p be the characteristic of the finite field GF(q), and let e be a divisor of q?1, e≥3. We determine the cyclotomic numbers of order e over GF(q) for the case where ?1 is a power of p modulo e. In this case most of the cyclotomic numbers are equal. We also prove a theorem about difference sets.     

Class numbers of definite binary quadratic lattices over algebraic function fields

Let k be an algebraic function field of one variable X having a finite field GF(q) of constants with q elements, q odd. Confined to imaginary quadratic extensions $\text{K}\text{k}$, class number formulas are developed for both the maximal and nonmaximal binary quadratic lattices L on (K, N), where N denotes the norm from K to k. The class numbers of L grow either with the genus g(k) of k (assuming the fields under consideration have bounded degree) or with the relative genus $\text{g(}\text{K}\text{k}\text{)}$ (assuming the lattices under consideration have bounded scale). In contrast to analogous theorems concerning positive definite binary quadratic lattices over totally real number fields, k is not necessarily totally real.