Denniston [12] constructed partial difference sets (PDS) with parameters (2(3m), (2 (m + r) -2(m) +2 (R))(2(m-1)),2(m)-2 (R)+(2(m+r)- 2(m+2r))(2(r-2)), (2(m+r)-2(m+2r))(2(r-1))) in elementary abelian groups of order 2 3 m for all m > 2 and 1 < r < m . These PDS arise from maximal arcs in the Desarguesian projective planes PG(2,2(m)). Davis et al. [10] and also De Winter [13] presented constructions of PDS with Denniston parameters ( p( 3 m) , ( p (m + r) - p(m) + p (R))(p(m - 1)) ,p (m) - p (R) + ( p (m + r) - p(m) + p (R))(p(r - 2)),(p(m+r) - p(m) + p (R))(p(r - 1))) in elementary abelian groups of order p 3 m for all m > 2 and r is an element of {1, m -1}, where p is an odd prime. The constructions in [10,13] are particularly intriguing, as it was shown by Ball, Blokhuis, and Mazzocca [1] that no nontrivial maximal arcs in PG(2, q( m) ) exist for any odd prime power q . In this paper, we show that PDS with Denniston parameters (q(3m), ( q (m + r) - q(m) + q (R))(q(m - 1)), q(m) - q (R) + ( q (m + r) - q(m) + q (R))(q(r -2)), ( q( m + r) - q(m )+ q (R))(q(r -1))) exist in elementary abelian groups of order q( 3 m) for all m > 2 and 1 < r < m , where q is an arbitrary prime power. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

Consider a simple, connected graph Γ with n vertices. Let C be a code of length n with its coordinates corresponding to the vertices of Γ. We define C as a storage code on Γ if, for any codeword c ∈ C , the information at each coordinate of c can be recovered by accessing its neighboring coordinates. The main problem here is to construct high-rate storage codes on triangle-free graphs. In this paper, we employ the polynomial method to address a question proposed by Barg and Zémor in 2022, demonstrating that the BCH family of storage codes on triangle-free Cayley graphs achieves a unit rate. Furthermore, we generalize the construction of the BCH family and obtain more storage codes of unit rate on triangle-free graphs. We also compare the BCH family with the other known constructions by examining the rate of convergence of 1/(1- R ( Cn )) with respect to the length n , where R ( Cn ) is the rate of code Cn . At last, we reveal a connection between the storage codes on triangle-free graphs and the Ramsey number R (3, t ), which leads to an upper bound for the rate of convergence of 1/(1 - R ( Cn )).

Let a and h be positive integers and let p be a prime. Let q1,..., qt be the distinct prime divisors of hand write Q(h) = {Sigma(t)(i) = 1c(i)q(i): c(i) is an element of Z, c(i) >= 0}. We provide constructions of group invariant Butson Hadamard matrices BH(G, h) in the following cases. 1. G =(Z(p))(2a) and at least one of the following conditions is satisfied. p(a) is an element of Q(h), p(a)+ 2 is an element of Q(h) and his even, p(a) + 1 = (q(1) - 1)(q(2) - 1) where q(1) and q(2) are distinct prime divisors of h. 2. G = Z(pa) x Z(pa) and p - 1, p is an element of Q(h). 3. G = (Z(p2))(a) and p(b) is an element of Q(h) for some divisor bof awith 1 <= b < a. 4. G = Px Z(pa) where Pis any abelian group of order p(a) and p is an element of Q(h). (C) 2021 Elsevier Inc. All rights reserved.

Finite flag-transitive affine planes have received much attention during the past fifty years because of their connections with other combinatorial objects such as spreads, planar functions, semifields and linearized polynomials. In 1964, Foulser completely determined the automorphism groups of finite flag-transitive affine planes. If a flag-transitive affine plane has a solvable automorphism group, then the affine plane is called solvable. The non-solvable flag-transitive affine planes have been completely classified in the 1990s. But the complete classification for the solvable case seems far out of reach. All known solvable flag-transitive affine planes can be classified into two types: $\\mathcal{C}$-planes and $\\mathcal{H}$-planes, where $\\mathcal{H}$-planes only occur in the odd characteristic case. In this paper, we construct the first flag-transitive affine plane of order $2^9$ over its kernel $\\bF_{2^3}$, which is not of type $\\mathcal{C}$ and the largest Singer subgroup of the translation complement has order $(2^3-1)(2^9+1)/9$.

In this paper, we construct two infinite families of tight sets with parameters (q^2r-2-1) and (q^2r-1-q^2r-2), respectively, in the Hermitian polar space ℋ(2r-1,q^2) for any r≥ 2 and any prime power q. Both families admit (q-1).(r,q^2).2.2e as the full automorphism group, where q=p^e, p is a prime, and e a positive integer.

Cameron-Liebler line classes were introduced in , and motivated by a question about orbits of collineation groups of (3,q). These line classes have appeared in different contexts under disguised names such as Boolean degree one functions, regular codes of covering radius one, and tight sets. In this paper we construct an infinite family of Cameron-Liebler line classes in (3,q) with new parameter x=(q+1)^2/3 for all prime powers q congruent to 2 modulo 3. The examples obtained when q is an odd power of two represent the first infinite family of Cameron-Liebler line classes in (3,q), q even.

We revisit the problem of constructing Menon-Hadamard difference sets. In 1997, Wilson and Xiang gave a general framework for constructing Menon-Hadamard difference sets by using a combination of a spread and four projective sets of type Q in PG ( 3 , q ). They also found examples of suitable spreads and projective sets of type Q for q = 5 , 13 , 17. Subsequently, Chen (1997) succeeded in finding a spread and four projective sets of type Q in PG ( 3 , q ) satisfying the conditions in the Wilson-Xiang construction for all odd prime powers q . Thus, he showed that there exists a Menon-Hadamard difference set in groups of order 4 q 4 for all odd prime powers q . However, the projective sets of type Q found by Chen have automorphisms different from those of the examples constructed by Wilson and Xiang. In this paper, we first generalize Chen's construction of projective sets of type Q by using “semi-primitive” cyclotomic classes. This demonstrates that the construction of projective sets of type Q satisfying the conditions in the Wilson-Xiang construction is much more flexible than originally thought. Secondly, we give a new construction of spreads and projective sets of type Q in PG ( 3 , q ) for all odd prime powers q , which generalizes the examples found by Wilson and Xiang. This solves a problem left open in Section 5 of the Wilson-Xiang paper from 1997.

In this paper, we survey constructions of and nonexistence results on combinatorial/ geometric structures which arise from unions of cyclotomic classes of finite fields. In particular, we survey both classical and recent results on difference sets related to cyclotomy, and cyclotomic constructions of sequences with low correlation. We also give an extensive survey of recent results on constructions of strongly regular Cayley graphs and related geometric substructures such as m-ovoids and i-tight sets in classical polar spaces.

In this paper, we construct an infinite family of hemisystems of the Hermitian surface H(3, q (2)). In particular, we show that for every odd prime power q congruent to 3 modulo 4, there exists a hemisystem of H(3, q (2)) admitting C(q3+1)/4: C-3.

In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis-Whiteman (1972). In particular, we show that there exists a skew Hadamard difference family with 2^u-1 blocks in the additive group of the finite field of order q^e for any prime power q≡ 2^u+1 (mod 2^u+1) with u≥ 2 and any positive integer e. In the aforementioned work of Szekeres, Whiteman, and Wallis-Whiteman, the constructions of skew Hadamard difference families with 2^u-1 (u=2 or 3) blocks in (𝔽_q^e,+) depend on the exponent e, with e≡ 1,2, or 3 (mod 4) when u=2, and e≡ 1 (mod 2) when u=3, respectively. Our more general construction, in particular, removes the dependence on e. As a consequence, we obtain new infinite families of skew Hadamard matrices.

In this paper, we give an algebraic construction of a new infinite family of Cameron–Liebler line classes with parameter x=q2−12 for q≡5 or 9(mod12), which generalizes the examples found by Rodgers in [26] through a computer search. Furthermore, in the case where q is an even power of 3, we construct the first infinite family of affine two-intersection sets in AG(2,q), which is closely related to our Cameron–Liebler line classes.

In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of Tr(x d ) over \(\mathbb{F}_{2^m }\) in the case where m = 2t, d = 3+2 t+1 and gcd(d, 2 m − 1) = 1.

We give a construction of strongly regular Cayley graphs on finite fields $\mathbb{F}_{q}$ by using union of cyclotomic classes and index 4 Gauss sums. In particular, we obtain two infinite families of strongly regular graphs with new parameters.

We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields by using union of cyclotomic classes of F"q of order N=2p"1^m, where p"1 is a prime and m a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.

We construct twelve infinite families of pseudocyclic and non-amorphic association schemes, in which each nontrivial relation is a strongly regular graph. Three of the twelve families generalize the counterexamples to A. V. Ivanov’s conjecture by Ikuta and Munemasa (Eur J Combin 31:1513–1519, 2010 ).

Let K be the finite field of order qm+1, which is regarded as an (m+1)-dimensional vector space over Fq. For each h-dimensional Fq-subspace V of K, α∈K and 0⩽t⩽qm+1−1, we define St(V,α)=∑v∈V(α+v)t. For each 1⩽h⩽m, we obtain sufficient conditions on t for the vanishing of St(V,α); when h=m, combining this result with some p-rank results from coding theory, we obtain necessary and sufficient conditions on t for the vanishing of St(V,α).

We give two constructions of strongly regular Cayley graphs on finite fields Fq by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new parameters.