  A strong $\ell$-ification of a matrix polynomial $P(\lambda)=\sum
A_i\lambda^i$ of degree $d$ is a matrix polynomial $\mathcal{L}(\lambda)$ of
degree $\ell$ having the same finite and infinite elementary divisors, and the
same numbers of left and right minimal indices as $P(\lambda)$. Strong
$\ell$-ifications can be used to transform the polynomial eigenvalue problem
associated with $P(\lambda)$ into an equivalent polynomial eigenvalue problem
associated with a larger matrix polynomial $\mathcal{L}(\lambda)$ of lower
degree. Typically $\ell=1$ and, in this case, $\mathcal{L}(\lambda)$ receives
the name of strong linearization. However, there exist some situations, e.g.,
the preservation of algebraic structures, in which it is more convenient to
replace strong linearizations by other low degree matrix polynomials. In this
work, we investigate the eigenvalue conditioning of $\ell$-ifications from a
family of matrix polynomials recently identified and studied by Dopico, P\'erez
and Van Dooren, the so-called block Kronecker companion forms. We compare the
conditioning of these $\ell$-ifications with that of the matrix polynomial
$P(\lambda)$, and show that they are about as well conditioned as the original
polynomial, provided we scale $P(\lambda)$ so that $\max\{\|A_i\|_2\}=1$, and
the quantity $\min\{\|A_0\|_2,\|A_d\|_2\}$ is not too small. Moreover, under
the scaling assumption $\max\{\|A_i\|_2\}=1$, we show that any block Kronecker
companion form, regardless of its degree or block structure, is about as
well-conditioned as the well-known Frobenius companion forms. Our theory is
illustrated by numerical examples.
