Planar functions and related group algebras

Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles

“…A bent function is said to be planar if |H| = |N|. Planar functions are extensively studied in [15,16,21,22,25]. When N is a trivial H -module, such a function f is also known as a perfect non-linear function.…”

Relative (pn,p,pn,n)-difference sets with GCD(p,n)=1

“…If there is an Abelian (5n, 5, 5n, n) RDS in G with (5, n) = 1, n > 1, then there is no integer j such that 5 j ≡ −1 mod exp(G) 5 , the largest divisor of exp(G) which is coprime with 5.…”

“…In this present paper, we study Abelian relative (pn, p, pn, n) difference sets with p an odd prime not dividing n through a group ring approach. In [9], the authors showed that there is no Abelian (pq, q, pq, p) RDS in Z p × Z 2 q with p, q being two distinct odd primes such that p > q, extending the result in [5]. In [7], the authors showed that there is no Abelian (3pq, 3, 3pq, pq) RDS in Z 2 3 × Z p × Z q with p, q being two distinct primes larger than 3.…”

The combinatorics of binary arrays

“…Hence, granting the conjecture on finite projective plane, the conjecture on ideal matrix will be settled in the affirmative. Hiramine (1992) proved (in a setting which is somewhat more general) that if there exists a planar function on Z 3p (p a prime number), then p < 5. Hence there does not exist a 3p × 3p ideal matrix…”