In 2002, P. Gaborit introduced two constructions of self-dual codes using quadratic residues, so-called pure and bordered construction, as a generalization of the Pless symmetry codes.
In this paper, ...we further study conditions under which the pure and the bordered construction using Paley designs and Paley graphs yield self-dual codes.
Special attention is given to the binary and ternary codes.
Further, we construct \(t\)-designs from supports of the codewords of a particular weight in the binary and ternary codes obtained.
Finding a reasonably good upper bound for the clique number of Paley graphs is an open problem in additive combinatorics. A recent breakthrough by Hanson and Petridis using Stepanov's method gives an ...improved upper bound on Paley graphs defined on a prime field Fp, where p≡1(mod4). We extend their idea to the finite field Fq, where q=p2s+1 for a prime p≡1(mod4) and a non-negative integer s. We show the clique number of the Paley graph over Fp2s+1 is at most min(ps⌈p2⌉,q2+ps+14+2p32ps−1).
Let q be an odd prime power. Denote by r(q) the value of q modulo 4. In this paper, we establish a linear fractional correspondence between two types of maximal cliques of size q+r(q)2 in the Paley ...graph of order q2.
The celebrated Erdős–Ko–Rado (EKR) theorem for Paley graphs of square order states that all maximum cliques are canonical in the sense that each maximum clique arises from the subfield construction. ...Recently, Asgarli and Yip extended this result to Peisert graphs and other Cayley graphs which are Peisert-type graphs with nice algebraic properties on the connection set. On the other hand, there are Peisert-type graphs for which the EKR theorem fails to hold. In this article, we show that the EKR theorem of Paley graphs extends to almost all pseudo-Paley graphs of Peisert-type. Furthermore, we establish the stability results of the same flavor.
A Diophantine m-tuple with property D(n), where n is a non-zero integer, is a set of m positive integers {a1,...,am} such that aiaj+n is a perfect square for all 1⩽i<j⩽m. It is known that ...Mn=sup{|S|:S is a D(n) m-tuple} exists and is O(log|n|). In this paper, we show that the Paley graph conjecture implies that the upper bound can be improved to ≪(log|n|)ϵ, for any ϵ>0.
Let GP (q2,m) be the m-Paley graph defined on the finite field with order q2. We study eigenfunctions and maximal cliques in generalised Paley graphs GP (q2,m), where m|(q+1). In particular, we ...explicitly construct maximal cliques of size q+1m or q+1m+1 in GP (q2,m), and show the weight-distribution bound on the cardinality of the support of an eigenfunction is tight for the smallest eigenvalue −q+1m of GP (q2,m). These new results extend the work of Baker et al. and Goryainov et al. on Paley graphs of square order. We also study the stability of the Erdős-Ko-Rado theorem for GP (q2,m) (first proved by Sziklai).
We describe a new class of maximal cliques, with a vector space structure, of Cayley graphs defined on the additive group of a field. In particular, we show that in the cubic Paley graph with order
q
...3
, the subfield with
q
elements forms a maximal clique. Similar statements also hold for quadruple Paley graphs and Peisert graphs with quartic order.
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