Let
$P\in \mathbb{F}_{2}[z]$
be such that
$P(0)=1$
and degree
$(P)\geq 1$
. Nicolas et al. [‘On the parity of additive representation functions’, J. Number Theory73 (1998), 292–317] proved that there exists a unique subset
${\mathcal{A}}={\mathcal{A}}(P)$
of
$\mathbb{N}$
such that
$\sum _{n\geq 0}p({\mathcal{A}},n)z^{n}\equiv P(z)~\text{mod}\,2$
, where
$p({\mathcal{A}},n)$
is the number of partitions of
$n$
with parts in
${\mathcal{A}}$
. Let
$m$
be an odd positive integer and let
${\it\chi}({\mathcal{A}},.)$
be the characteristic function of the set
${\mathcal{A}}$
. Finding the elements of the set
${\mathcal{A}}$
of the form
$2^{k}m$
,
$k\geq 0$
, is closely related to the
$2$
-adic integer
$S({\mathcal{A}},m)={\it\chi}({\mathcal{A}},m)+2{\it\chi}({\mathcal{A}},2m)+4{\it\chi}({\mathcal{A}},4m)+\cdots =\sum _{k=0}^{\infty }2^{k}{\it\chi}({\mathcal{A}},2^{k}m)$
, which has been shown to be an algebraic number. Let
$G_{m}$
be the minimal polynomial of
$S({\mathcal{A}},m)$
. In precedent works there were treated the case
$P$
irreducible of odd prime order
$p$
. In this setting, taking
$p=1+ef$
, where
$f$
is the order of
$2$
modulo
$p$
, explicit determinations of the coefficients of
$G_{m}$
have been made for
$e=2$
and 3. In this paper, we treat the case
$e=4$
and use the cyclotomic numbers to make explicit
$G_{m}$
.