New Necessary Conditions for Existence of Strong External Difference Families
Jingjun Bao, Lijun Ji
TL;DR
The paper tackles the existence problem for strong external difference families by merging cyclotomic-field methods with algebraic number theory and character theory. It introduces two new exponent-bounds for the ambient abelian group via prime-ideal decomposition and Schmidt's field descent, and it develops a field-descent framework using Gauss sums to obtain prime-divisor and congruence constraints on the parameters for $m>2$. Additionally, Schmidt’s norm-theory provides a complementary bound that reinforces nonexistence results for various parameter sets. Collectively, these results extend known nonexistence criteria and deepen the theoretical understanding of SEDFs, with implications for AMD-code constructions and related combinatorial designs.
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 $λ$ for $(v,m,k,λ)$-SEDFs with $m>2$ are established.