Completeness of Normal Rational Curves

The completeness of normal rational curves, considered as (q + 1)-arcs in PG(n, q), is investigated. Previous results of Storme and Thas are improved by using a result by Kovács. This solves the problem completely for large prime numbers q and odd nonsquare prime powers q = p ^2h+1 with p prime, , where p ₀(h) is an odd prime number which depends on h.     

On the number of lines in planar spaces

SupposeS is a planar space withv>4 points and letq be the positive real number such thatv=q ³+q²+q+1. Assuming a weak non-degeneracy condition, we shall show thatS has at least (q²+1)(q²+q+1) lines with equality iffq is a prime power andS=PG(3,q).     

On the number of lines in 4-dimensional linear spaces

Assuming a weak non-degeneracy condition, we show that a linear spaceL of dimension at least 4 withv=q ⁴+q ³+q ²+q+1 points,q > 1 any positive real number, has at least (q²+1)v lines with equality if and only ifq is a prime power andL = PG(4,q).Dedicated to H. Mäurer on the occasion of his 60th birthday     

An upper bound for the minimum weight of dual codes of Figueroa planes

In this note we give an explicit construction for words of weight 2q³ - q² - q in the dual p-ary code of the Figueroa plane of order q³, where q > 2 is any power of the prime p. When p is odd this then allows us, for the Figueroa planes, to improve on the previously known upper bound of 2q³ for the minimum weight of the dual p-ary code of any plane of order q³. The construction is the same as one that applies to desarguesian planes of order q³ as described in [3].     

Improving two theorems of bose on difference families

In [2] R. C. Bose gives a sufficient condition for the existence of a (q, 5, 1) difference family in (GF(q), +)—where q ≡ 1 mod 20 is a prime power — with the property that every base block is a coset of the 5th roots of unity. Similarly he gives a sufficient condition for the existence of a (q, 4, 1) difference family in (GF(q, +)—where q ≡ 1 mod 12 is a prime power — with the property that every base block is the union of a coset of the 3rd roots of unity with zero. In this article we replace the mentioned sufficient conditions with necessary and sufficient ones. As a consequence, we obtain new infinite classes of simple difference families and hence new Steiner 2-designs with block sizes 4 and 5. In particular, we get a (p^4α, 5, 1)-DF for any odd prime p ≡ 2, 3 (mod 5), and a (p^2α, 4, 1)-DF for any odd prime p ≡ 2 (mod 3). © 1995 John Wiley & Sons, Inc.     

A remark on the uniqueness of embeddings of linear spaces into desarguesian projective planes

Suppose that q ? 2 is a prime power. We show that a linear space with a(q + 1)² + (q + 1) points, where a ? 0.763, can be embedded in at most one way in a desarguesian projective plane of order q. © 1995 John Wiley & Sons, Inc.     

On the existence of semifields of prime power order admitting free automorphism groups

The purpose of this paper is to prove the existence of semifields of order q ⁴ for any odd prime power q = p^r, q > 3, admitting a free automorphism group isomorphic to Z ₂ × Z ₂.     

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.     

Some new symmetric designs

For q, an odd prime power, we construct symmetric (2q²+2q+1,q²,½q(q-1)) designs having an automorphism group of order q that fixes 2q+1 points. The construction indicates that for each q the number of such designs that are pairwise non-isomorphic is very large.     

The Existence of Four HMOLS with Equal Sized Holes

In this paper, we present several new constructions for k holey mutually orthogonal Latin squares (HMOLS) of type g ⁿ . We concentrate mainly on k=4; here, for all but two values of n, namely 6 and 15, only a finite number of unsolved cases remain. Some new sets of 5 and 6 HMOLS are also given, in particular 5 HMOLS(2 ^q ) for q63 or q an odd prime power between 6 and 62, plus 6 HMOLS(4 ^q )for q an odd prime power between 8 and 60.     

On the least prime in an arithmetic progression

Letq, k be positive integers, (q, k)=1, andP(k, q) be the smallest prime number satisfyingpэk (modq) In this paper, we prove that forq sufficiently large, one hasP (k,q)≪q ⁸.     

Multiple prime covers of the riemann sphere

A compact Riemann surface X of genus g≥2 which admits a cyclic group of automorphisms C _q of prime order q such that X/C _q has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p≠q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p.     

On completely free elements in finite fields

Letq>1 be a prime power,m>1 an integer,GF(q ^m) andGF (q) the Galois fields of orderq ^m andq, respectively. We show that the different module structures of (GF(q ^m), +) arising from the intermediate fields of the field extensionGF(q ^m) overGF (q), can be studied simultaneously with the help of some basic properties of cyclotomic polynomials. The results can be generalized to finite cyclic Galois extensions over arbitrary fields.In 1986, D. Blessenohl and K. Johnsen proved that there exist elements inGF(q ^m) which generate normal bases inGF(q ^m) overany intermediate fieldGF(q ^d) ofGF(q ^m) overGF(q). Such elements are called completely free inGF(q ^m) overGF(q). Using our ideas, we give a detailed and constructive proof of the most difficult part of that theorem, i.e., the existence of completely free elements inGF(q ^m), overGF(q) provided thatm is a prime power. The general existence problem of completely free elements is easily reduced to this special case.Furthermore, we develop a recursive formula for the number of completely free elements inGF(q ^m) overGF(q) in the case wherem is a prime power.     

Simple quasidoubles of projective planes

We show that the number of (simple) designsS ₂(2,q + 1,q ² +q + 1), whereq is a power of a prime, grows at least asO(q!).Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Constructions of partitions of FqN into perfect q-ary codes

Given N = (q ^m − 1)/(q − 1), where q is a power of a prime, q > 2, we present two constructions of different partitions of the set F _q ^N of all q-ary length N vectors into perfect q-ary codes of length N. The lower bounds on the number of these partitions are presented.     

On the generators of subgroups of unit groups of group rings

In this paper we find the generators of a subgroup of finite index in the unit group of the integral group ring of the metacyclic group of orderpq given byG=(a,x:a ^p=1=x ^q ,xax ⁻¹=a ^f ), wherep is an odd prime,q>2 a divisor ofp-1, andf belongs to the exponentq modulop.     

The weight distributions of irreducible cyclic codes of length

Let m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2^m over F_q are determined explicitly.     

Structure Theorems for Semisimple Hopf Algebras of Dimension pq 3

Let p, q be prime numbers with p > q ³, and k an algebraically closed field of characteristic 0. In this article, we obtain the structure theorems for semisimple Hopf algebras of dimension pq ³.     

The Insecurity of the Elliptic Curve Digital Signature Algorithm with Partially Known Nonces

Nguyen and Shparlinski have recently presented a polynomial-time algorithm that provably recovers the signer's secret DSA key when a few consecutive bits of the random nonces k (used at each signature generation) are known for a number of DSA signatures at most linear in log q (q denoting as usual the small prime of DSA), under a reasonable assumption on the hash function used in DSA. The number of required bits is about log ^1/2 q, but can be decreased to log log q with a running time q ^{O(1/log log q)} subexponential in log q, and even further to two in polynomial time if one assumes access to ideal lattice basis reduction, namely an oracle for the lattice closest vector problem for the infinity norm. All previously known results were only heuristic, including those of Howgrave-Graham and Smart who introduced the topic. Here, we obtain similar results for the elliptic curve variant of DSA (ECDSA).     

On a Decomposition of Complete Graphs

It is shown, among other results, that for any prime power q, the complete graph on 1+q+q ²+q ³ vertices can be decomposed into a union of 1+q Siamese Strongly Regular Graphs S R G(1+q+q ²+q ³,q+q ²,q–1,q+1) sharing 1+q ² cliques of size 1+q. Acknowledgments.The authors are indebted to a referee for a very extensive report and for many suggestions which improved the presentation of the paper tremendously.AMS Subject Numbers: 05B05, 05B20, 05E30This work was completed while the first author was on sabbatical leave visiting Institute for studies in theoretical Physics and Mathematics, (IPM), in Tehran, Iran. Support and hospitality is appreciated. Supported by an NSERC operating grant.