  A $(t,s,v)$-all-or-nothing transform (AONT) is a bijective mapping defined on
$s$-tuples over an alphabet of size $v$, which satisfies that if any $s-t$ of
the $s$ outputs are given, then the values of any $t$ inputs are completely
undetermined. When $t$ and $v$ are fixed, to determine the maximum integer $s$
such that a $(t,s,v)$-AONT exists is the main research objective. In this
paper, we solve three open problems proposed in [IEEE Trans. Inform. Theory 64
(2018), 3136-3143.] and show that there do exist linear $(2,p,p)$-AONTs. Then
for the size of the alphabet being a prime power, we give the first infinite
class of linear AONTs which is better than the linear AONTs defined by Cauchy
matrices. Besides, we also present a recursive construction for general AONTs
and a new relationship between AONTs and orthogonal arrays.
