A lower bound on the field size of convolutional codes with a maximum distance profile and an improved construction
Zitan Chen
TL;DR
This work advances the theory of MD P convolutional codes by (i) proving the first nontrivial asymptotic lower bound on the field size required to construct $(n,k,\delta)$ MD P codes with maximum profile length $L$, showing $q = \Omega_L(n^{L-1})$, and (ii) providing an explicit L=1 MD P construction over a field of size $\Theta(n^{\delta})$ (with $\delta=\min\{k,n-k\}$) based on skew polynomials and skew Vandermonde matrices. The lower bound rules out MD P constructions with $L\ge3$ over $O(n)$-sized fields for general parameters, while the L=1 construction improves upon prior field-size bounds $O(n^{2\delta})$ and yields practical explicit codes via a skew-polynomial framework. The combination of a probabilistic field-size bound and an explicit algebraic construction offers new insight into the trade-off between code optimality (MDP) and the required field size, with direct implications for streaming error correction. The results illuminate both the limits and possibilities for polynomial-in-$n$ field sizes in MD P code design, and lay groundwork for further reductions toward tighter constructions for broader parameter regimes.
Abstract
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 $Ω_L(n^{L-1})$ 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 >= 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.