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 <inline-formula> <tex-math notation="LaTeX">$k/n$ </tex-math></inline-formula> and a maximum distance profile require a field of size at least exponential in <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. In this paper, we show that a finite field of size <inline-formula> <tex-math notation="LaTeX">$\Omega _{L}(n^{L-1})$ </tex-math></inline-formula> is necessary for constructing convolutional codes with rate <inline-formula> <tex-math notation="LaTeX">$k/n$ </tex-math></inline-formula> and a maximum distance profile of length <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>. As a direct consequence, this rules out the possibility of constructing convolutional codes with a maximum distance profile of length <inline-formula> <tex-math notation="LaTeX">$L\geq 3$ </tex-math></inline-formula> over a finite field of size <inline-formula> <tex-math notation="LaTeX">$O(n)$ </tex-math></inline-formula>. Additionally, we also present an explicit construction of convolutional code with rate <inline-formula> <tex-math notation="LaTeX">$k/n$ </tex-math></inline-formula> and a maximum profile of length <inline-formula> <tex-math notation="LaTeX">$L=1$ </tex-math></inline-formula> over a finite field of size <inline-formula> <tex-math notation="LaTeX">$O(n^{\min \{k,n-k\}})$ </tex-math></inline-formula>, achieving a smaller field size than known constructions with the same profile length.