New existence and nonexistence results for strong external difference families

Abstract

In this paper, we use character-theoretic techniques to give new nonexistence results for (n, m, k, λ)strong external difference families (SEDFs). We also use cyclotomic classes to give two new classes of SEDFs with m = 2. IntroductionMotivated by applications to algebraic manipulation detection codes (or AMD codes) [1, 2, 3], Paterson and Stinson introduced strong external difference families (or SEDFs) in [10]. SEDFs are closely related to but stronger than external difference families (or EDFs) [9]. In [10], it was noted that optimal AMD codes can be obtained from EDFs, whereas optimal strong AMD codes can be obtained from SEDFs. See [10] for a discussion of these and related structures and how they relate to AMD codes.Recently, Matin and Stinson [8] and Hucznska and Paterson [5] further investigated the existence of SEDFs and gave some new results about the nonexistence of certain SEDFs. Their results show that the existence of SEDFs is an interesting mathematical problem in its own right, independent of any applications to AMD codes. In [8], the authors presented some nonexistence results of SEDFs by using character theory. In this paper, we shall explore such a method to give some new nonexistence results for SEDFs. Let us recall the definition of SEDFs.Let G be a finite abelian group of order n (written multiplicatively) with the identity e ∈ G. For any two nonempty subsets A 1 , A 2 of G, the multisetis called the external difference of A 1 and A 2 . Let k, λ, m be positive integers and let A 1 , . . . , A m be (pairwise disjoint) k-subsets of G. If the following multiset equation holds: {ℓ : ℓ =j} ∆ E (A ℓ , A j ) = λ(G \ {e}) * Research supported by NSFC grant 11431003 (L. Ji), 11301457 (Y. Zhang), and NSERC RGPIN-16-05610 (R. Wei).

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)…”

“…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.…”

Some results on generalized strong external difference families

“…Wen, Yang, Fu and Feng [19] present some general constructions of GSEDF by using difference sets and partial difference sets. There are some (v, 2, k, λ)-SEDFs obtained from cyclotomic constructions, see [1,8,15]. Wen, Yang and Feng [18], and Jedwab and Li [9] respectively give an example of (243, 11,22,20)-SEDF in two different ways which is the first nontrivial example for m ≥ 5.…”

“…Further, Jedwab and Li [9] gave some upper bounds for a (v, m, k, λ)-SEDF, and used them to get some nonexistence results for the case m = 2. For more nonexistence results on (v, m, k, λ)-SEDFs, see [1,8,9,14].…”

“…There exists a(16, 2; 5, 9; 3, 3)-GSEDF.Proof. Let G = Z 2 ×Z 8 , D 1 = {(0, 0), (0, 1), (0, 3), (1, 0),(1,4)}, and D 2 = {(0, 4), (0, 5), (0, 7), (1, 1), (1, 2), (1, 3),(1,5),(1,6), (1, 7)}. It is easy to check that ∆(D 1 ,D 2 ) = ∆(D 2 , D 1 ) = (Z 2 × Z 8 ) \ {(0, 0)}.…”