From a 1-Rotational RBIBD to a Partitioned Difference Family

Abstract

Generalizing the case of $\lambda=1$ given by Buratti and Zuanni [Bull Belg. Math. Soc. (1998)], we characterize the $1$-rotational difference families generating a 1-rotational $(v,k,\lambda)$-RBIBD, that is a $(v,k,\lambda)$ resolvable balanced incomplete block design admitting an automorphism group $G$ acting sharply transitively on all but one point $\infty$ and leaving invariant a resolution $\cal R$ of it. When $G$ is transitive on $\cal R$ we prove that removing $\infty$ from a parallel class of $\cal R$ one gets a partitioned difference family, a concept recently introduced by Ding and Yin [IEEE Trans. Inform. Theory, 2005] and used to construct optimal constant composition codes. In this way, by exploiting old and new results about the existence of 1-rotational RBIBDs we are able to derive a great bulk of previously unnoticed partitioned difference families. Among our RBIBDs we construct, in particular, a $(45,5,2)$-RBIBD whose existence was previously in doubt.

On optimal weak algebraic manipulation detection codes and weighted external difference families

“…According to Lemma 2, B (1) is an optimal (10, 3, (1, 2, 2), 5, 3)-BSWEDF and B (2) is an optimal (10, 3, (1, 1, 3), 5, 4)-BSWEDF. By Corollary 1, ρ (10,3,5)…”

“…Example 1: Let n = 10, m = 3, and a = 5. Let B = {{5}, {2}, {0, 4, 6}} be a family of disjoint subsets of Z 10 , which corresponding to a weak (10,3,5…”

“…i.e., U is an (n = 3q, m = 4, (1, 1, 2k, 2k), a = 4k + 2, λ = 2k + 1)-BSWEDF. , 2 1 ), 2 , v = p m 1 1 p m 2 2 · · · p mr r , 2 < p 1 < p 2 < · · · < pr, and 3|(pt − 1) for 1 ≤ t ≤ r [3] sv, ((s + 1)…”

“…sv−s s+1 , s 1 ), s v = p m 1 1 p m 2 2 · · · p mr r , 2 < p 1 < p 2 < · · · < pr, and 2(s + 1)|(pt − 1) for 1 ≤ t ≤ r, s = 4, 5 [3] 6v, (7 6v− 6 7 , 6 1 ), 6 v = p m 1 1 p m 2 2 · · · p mr r , 2 < p 1 < p 2 < · · · < pr, and 28|(pt − 1) for 1 ≤ t ≤ r [3] 7v, (8…”

“…Firstly, we define a new type of weighted external difference families which are proved equivalent with weak AMD codes. By means of this combinatorial characterization of weak AMD codes: (1) We improve the known lower bound on the maximum probability of successful tampering for the adversary's all possible strategies; (2) We derive a necessary condition for the Paterson-Stinson bound to be achieved; (3) We determine the exact combinatorial structure for a weak AMD code to be optimal, when the Paterson-Stinson bound is not achievable. In this way, some weak AMD codes which have not been identified to be R-optimal previously now can be identified to be in fact R-optimal.…”