A Lower Bound on the Field Size of Convolutional Codes With a Maximum Distance Profile and an Improved Construction

Convolutional codes with a maximum distance profile attain the largest possible column distances for the maximum number of time instants and thus have outstanding error-correcting capability especially for streaming applications. Explicit constructions of such codes are scarce in the literature. In particular, known constructions of convolutional codes with rate
$k/n$
and a maximum distance profile require a field of size at least exponential in
$n$
for general code parameters. At the same time, the only known lower bound on the field size is the trivial bound that is linear in
$n$
. In this paper, we show that a finite field of size
$\Omega _{L}(n^{L-1})$
is necessary for constructing convolutional codes with rate
$k/n$
and a maximum distance profile of length
$L$
. As a direct consequence, this rules out the possibility of constructing convolutional codes with a maximum distance profile of length
$L\geq 3$
over a finite field of size
$O(n)$
. Additionally, we also present an explicit construction of convolutional code with rate
$k/n$
and a maximum profile of length
$L=1$
over a finite field of size
$O(n^{\min \{k,n-k\}})$
, achieving a smaller field size than known constructions with the same profile length.