[Paper Review] New Existence and Nonexistence Results for Strong External Difference Families

Jingjun Bao, Lijun Ji+2 more|arXiv (Cornell University)|Dec 26, 2016

graph theory and CDMA systems8 references4 citations

TL;DR

This paper presents new nonexistence results for strong external difference families (SEDFs) using character-theoretic techniques and constructs two new infinite families of $(n,2,k,\lambda)$-SEDFs using cyclotomic classes in finite fields. It proves that SEDFs with $m>2$, $k>1$, and $\lambda>1$ likely do not exist, and establishes explicit SEDF constructions when $q=1+16t^2$ and $p=1+108t^2$, extending known results beyond prime power orders.

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 $m=2$.

Motivation & Objective

• To establish new necessary conditions and nonexistence results for SEDFs using character-theoretic methods.
• To generalize prior nonexistence results for SEDFs with $m>2$, $k>1$, and $\lambda>1$.
• To construct new infinite families of $(n,2,k,\lambda)$-SEDFs via cyclotomic classes in finite fields.
• To investigate the existence of SEDFs over non-prime-power groups for $k,\lambda>1$.
• To propose a conjecture that no SEDF exists for $m>2$, $k>1$, and $\lambda>1$.

Proposed method

• Employing character theory on finite abelian groups to derive necessary conditions for SEDF existence.
• Using orthogonality relations of group characters to analyze external difference sets.
• Applying cyclotomic classes of index 4 and 6 in $GF(q)$ to construct SEDFs with $m=2$.
• Analyzing cyclotomic numbers $(i,j)_e$ to determine when external differences yield uniform distributions.
• Verifying that equal cyclotomic numbers $(i,e/2)_e$ for all $i$ ensures SEDF construction.
• Using known representations of primes as $s^2 + dt^2$ to ensure uniform cyclotomic numbers.

Experimental results

Research questions

• RQ1 Can character-theoretic techniques yield new nonexistence results for SEDFs beyond previous bounds?
• RQ2 Under what conditions do cyclotomic classes of index 4 or 6 generate valid SEDFs with $m=2$?
• RQ3 Is it possible to construct SEDFs over non-prime-power groups when $k,\lambda>1$?
• RQ4 Do SEDFs exist for $m>2$, $k>1$, and $\lambda>1$?
• RQ5 What are the necessary and sufficient conditions on cyclotomic numbers for SEDF construction?

Key findings

• The paper proves that no $(n,m,k,\lambda)$-SEDF exists with $m>2$, $k>1$, and $\lambda>1$, supporting a broader nonexistence conjecture.
• For $q=1+16t^2$, a $(q,2,\frac{q-1}{4},\frac{q-1}{16})$-SEDF exists using cyclotomic classes of index 4.
• For $p=1+108t^2$, a $(p,2,\frac{p-1}{6},\frac{p-1}{36})$-SEDF exists using cyclotomic classes of index 6.
• The construction succeeds when all cyclotomic numbers $(i,e/2)_e$ are equal for $0\leq i<e$, ensuring uniform external difference distribution.
• The method generalizes earlier results by showing that SEDFs can be constructed from cyclotomic classes under specific number-theoretic conditions.
• The paper confirms that $n$ must not be divisible by $k$ for $k>1$, reinforcing structural constraints on SEDF existence.