New Necessary Conditions for Existence of Strong External Difference Families

TL;DR

This paper establishes new necessary conditions for the existence of strong external difference families (SEDFs) using advanced algebraic number theory and character theory, providing bounds and relations that constrain their parameters.

Contribution

It introduces novel algebraic number theory techniques to derive necessary conditions and bounds for SEDFs, advancing theoretical understanding of their existence.

Findings

Derived exponent bounds for SEDFs.

Established congruence relations between parameters.

Provided bounds for prime divisors of group order.

Abstract

Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. In this paper, we use the theory of cyclotomic fields, algebraic number theory and character theory to give some new necessary conditions for the existence of SEDFs. Based on the results of decomposition of prime ideals and Schmidt's field descent method, two exponent bounds of SEDFs are presented. Based on the field descent method, a special homomorphism from an abelian group to its cyclic subgroup and Gauss sums, some bounds for prime divisors of $v$ and some congruence relations between $k, m$ and $\lambda$ for $(v,m,k,\lambda)$-SEDFs with $m>2$ are established.