Some results on generalized strong external difference families

A generalized strong external difference family (briefly $(v, m; k_1,\dots,k_m; \lambda_1,\dots,\lambda_m)$-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we construct some new GSEDFs for $m=2$ and use them to obtain some results on graph decomposition. We also give some nonexistence results for GSEDFs. Especially, we prove that a $(v, 3;k_1,k_2,k_3; \lambda_1,\lambda_2,\lambda_3)$-GSEDF does not exist when $k_1+k_2+k_3< v$.Comment: generalized strong external difference family; difference set; character theory; graph decomposition; nonexistenc

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

This paper provides a combinatorial characterization of weak algebraic manipulation detection (AMD) codes via a kind of generalized external difference families called bounded standard weighted external difference families (BSWEDFs). By means of this characterization, we improve a known lower bound on the maximum probability of successful tampering for the adversary's all possible strategies in weak AMD codes. We clarify the relationship between weak AMD codes and BSWEDFs with various properties. We also propose several explicit constructions for BSWEDFs, some of which can generate new optimal weak AMD codes

Algebraic Manipulation Detection Codes via Highly Nonlinear Functions

In this paper, we study the relationship between algebraic manipulation detection (AMD) codes and highly nonlinear functions. As applications, on one hand, a generic construction for systematic AMD codes is introduced based on highly nonlinear functions. Systematic AMD codes with new parameters can be generated from known highly nonlinear functions. Especially, several infinite classes of optimal systematic AMD codes, some with asymptotically optimal tag size, can be constructed. On the other hand, systematic AMD codes are used to construct highly nonlinear functions. The known construction by Cramer et al. [10] for systematic AMD codes turns out to be based on a special kind of functions with high nonlinearity

Nonexistence of Strong External Difference Families in Abelian Groups of Order Being Product of At Most Three Primes

Let $v$ be a product of at most three not necessarily distinct primes. We prove that there exists no strong external difference family with more than two subsets in abelian group $G$ of order $v$, except possibly when $G=C_p^3$ and $p$ is a prime greater than $3 \times 10^{12}$.Comment: 30 pages, major changes made to version 1, we refined proofs to some theorems and reorganized the exposition of many result