Improved Bounds for (b, k)-Hashing

For fixed integers
$n$
and
$b\geq k$
, let
$A(b,k,n)$
the largest size of a subset of
$\{1,2,\ldots,b\}^{n}$
such that, for any
$k$
distinct elements in the set, there is a coordinate where they all differ. Bounding
$A(b,k,n)$
is a problem of relevant interest in information theory and computer science, relating to the zero-error capacity with list decoding and to the study of
$(b, k)$
-hash families of functions. It is known that, for fixed
$b$
and
$k$
,
$A(b,k,n)$
grows exponentially in
$n$
. In this paper, we determine new exponential upper bounds for different values of
$b$
and
$k$
. A first bound on
$A(b,k,n)$
for general
$b$
and
$k$
was derived by Fredman and Komlós in the ’80s and improved for certain
$b\neq k$
by Körner and Marton and by Arikan. Only very recently better bounds were derived for general
$b$
and
$k$
by Guruswami and Riazanov, while stronger results for small values of
$b=k$
were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan, and by Costa and Dalai. In this paper, we strengthen the bounds for some specific values of
$b$
and
$k$
. Our contribution is a new computational method for obtaining upper bounds on the values of a quadratic form defined over discrete probability distributions in arbitrary dimensions, which emerged as a central ingredient in recent works. The proposed method reduces an infinite-dimensional problem to a finite one, which we manage to further simplify by means of a series of optimality conditions.