On the distribution of points in orbits of PGL(2,q) acting on GF(q)

We prove results on the distribution of points in an orbit of PGL(2,q) acting on an element of GF(qⁿ). These results support a conjecture of Klapper. More precisely, we show that the points in an orbit are uniformly distributed if n is small with respect to q.     

Linear (q+1)-fold Blocking Sets in PG(2, q)

A (q+1)-fold blocking set of size (q+1)(q⁴+q²+1) in PG(2, q⁴) which is not the union of q+1 disjoint Baer subplanes, is constructed     

A four-dimensional Kerdock set over GF(3)

If m is an even integer and K = GF(q) is a field of characteristic 2, then there exists a set of q^m?1 alternating bilinear forms of degree m over K such that the difference of any two of the forms is nonsingular. Do such sets exist over fields of odd characteristic? This note constructs such a set in the smallest nontrivial case, namely, m = 4, q = 3.     

On the k-error linear complexity of sequences with period 2pn over GF(q)

In this paper, we first optimize the structure of the Wei–Xiao–Chen algorithm for the linear complexity of sequences over GF(q) with period N =  2p ⁿ , where p and q are odd primes, and q is a primitive root modulo p ². The second, an union cost is proposed, so that an efficient algorithm for computing the k-error linear complexity of a sequence with period 2p ⁿ over GF(q) is derived, where p and q are odd primes, and q is a primitive root modulo p ². The third, we give a validity of the proposed algorithm, and also prove that there exists an error sequence e ^N , where the Hamming weight of e ^N is not greater than k, such that the linear complexity of (s + e) ^N reaches the k-error linear complexity c. We also present a numerical example to illustrate the algorithm. Finally, we present the minimum value k for which the k-error linear complexity is strictly less than the linear complexity.     

Subregular Spreads of (2n+1,q)

In this paper, we develop some of the theory of spreads of projective spaces with an eye towards generalizing the results of R. H. Bruck (1969,in“Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514, Univ. of North Carolina Press, Chapel Hill). In particular, we wish to generalize the notion of asubregularspread to the higher dimensional case. Most of the theory here was anticipated by Bruck in later papers; however, he never provided a detailed formulation. We fill this gap here by developing the connections between a regular spread of (2n+1)-dimensional projective space and ann-dimensional circle geometry, which is the appropriate generalization of the Miquelian inversive plane. After developing this theory, we provide a fairly general method for constructing subregular spreads of (5,q). Finally, we explore a special case of this construction, which yields several examples of three-dimensional subregular translation planes which are not André planes.     

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.     

Matroids with No (q+2)-Point-Line Minors

It is known that a geometry with rankrand no minor isomorphic to the (q+2)-point line has at most (q^r−1)/(q−1) points, with strictly fewer points ifr>3 andqis not a prime power. Forqnot a prime power andr>3, we show thatq^r−1−1 is an upper bound. Forqa prime power andr>3, we show that any rank-rgeometry with at leastq^r−1points and no (q+2)-point-line minor is representable overGF(q). We strengthen these bounds toq^r−1−(q^r−2−1)/(q−1)−1 andq^r−1−(q^r−2−1)/(q−1) respectively whenqis odd. We give an application to unique representability and a new proof of Tutte's theorem: A matroid is binary if and only if the 4-point line is not a minor.     

A characterization of PSL(3,q) where q is an odd prime power

Order components of a finite group are introduced in Chen (J. Algebra 15 (1996) 184). We prove that PSL(3,q) can be uniquely determined by its order components where q is an odd prime power. A main consequence of our result is the validity of Thompson's conjecture for the groups under consideration.     

The SU(v) Calogero spin system

A one-dimensional quantum particle system in which particles with su(v) spins interact through inverse square interactions is introduced. We refer to it as the SU(v) Calogero spin system. Using the quantum inverse scattering method, we reveal algebraic structures of the system: hidden symmetry is the U(v) − SU(v) U(1) current algebra. This is consistent with the fact that the ground-state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, known as the w_{1 + ∞}-algebra. With this W-algebra we have a unified viewpoint on the integrable quantum particle systems with long-range interactions such as the Calogero type (1/x²-interactions) and Sutherland type (1/sin²x-interactions). The Yangian symmetry is briefly discussed.     

The character table of a split extension of the Heisenberg group H 1( q ) by Sp (2, q ), q odd

In this paper we determine the full character table of a certain split extension of the Heisenberg group H ₁ by the odd-characteristic symplectic group Sp(2, q).      

Mixed partitions of PG(3,q)

A mixed partition of PG(2n−1,q²) is a partition of the points of PG(2n−1,q²) into (n−1)-spaces and Baer subspaces of dimension 2n−1. In (Bruck and Bose, J. Algebra 1 (1964) 85) it is shown that such a mixed partition of PG(2n−1,q²) can be used to construct a (2n−1)-spread of PG(4n−1,q) and hence a translation plane of order q²ⁿ. In this paper, we provide several new examples of such mixed partitions in the case when n=2.     

Hopf *-algebra structures on H(1, q)

We study the Hopf *-algebra structures on the Hopf algebra H(1, q) over ?. It is shown that H(1, q) is a Hopf *-algebra if and only if |q| = 1 or q is a real number. Then the Hopf *-algebra structures on H(1, q) are classified up to the equivalence of Hopf *-algebra structures.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

Decompositions of vector spaces over GF(2) into disjoint equidimensional affine spaces

We investigate the possibilities for decomposing the vector space [GF(2)]ⁿ into a set of 2^r?d (necessarily disjoint) d dimensional affine subspaces.     

On irreducible p,q-representations of gl(2)

The theory of irreducible p,q-representations of the complex Lie algebra gl(2) is developed. We construct a one variable model of irreducible p,q-representations of gl(2) in terms of p,q-derivative operator, and derive a generating function based on it.     

The invariant polynomials characterizing U(n) tensor operators p, q,…, q, 0,…, 0 having maximal null space

We make use of the “path sum” function to prove that the family of stretched operator functions characterized by the operator irrep labels p,q,…,q, 0,…, 0 satisfy a pair of general difference equations. This family of functions is a generalization of Milne's p,q,…,q, 0, functions for U(n) and Biedenharn and Louck's p,q, 0 functions for U(3). The fact that this family of stretched operator functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations and symmetry properties we give an explicit formula for the polynomials characterized by the operator irrep labels p, 1, 0,…, 0.     

Fast algorithms for determining the linear complexities of sequences over GF(pm) with the period 3n

In this paper, for the the primes p such that 3 is a divisor of p − 1, we prove a result which reduces the computation of the linear complexity of a sequence over GF(p ^m) (any positive integer m) with the period 3n (n and p ^m − 1 are coprime) to the computation of the linear complexities of three sequences with the period n. Combined with some known algorithms such as generalized Games-Chan algorithm, Berlekamp-Massey algorithm and Xiao-Wei-Lam-Imamura algorithm, we can determine the linear complexity of any sequence over GF(p ^m) with the period 3n (n and p ^m − 1 are coprime) more efficiently.     

Constructions of (q, K, λ, t, Q) almost difference families

The concept of a (q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, (q, K, λ, t, Q)-ADFs, where K = {k₁, k₂, ..., k_r} is a set of positive integers and Q = (q₁, q₂,..., q_r) is a given block-size distribution sequence. A necessary condition for the existence of a (q, K, λ, t, Q)-ADF is given, and several infinite classes of (q, K, λ, t, Q)-ADFs are constructed.     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.