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 >= 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.

翻译：具有最大距离轮廓的卷积码能在最多时间瞬间上达到最大的列距离，因此在流式传输等应用中具有卓越的纠错能力。文献中此类码的显式构造十分罕见。特别地，对于一般码参数，速率k/n且具有最大距离轮廓的卷积码的已知构造需要域大小至少关于n呈指数增长。同时，已知的唯一域大小下界是平凡线性界。本文证明，构造长度为L、速率k/n且具有最大距离轮廓的卷积码需要大小为$\Omega_L(n^{L-1})$的有限域。这一结论直接排除了在大小为O(n)的有限域上构造长度L>=3且具有最大距离轮廓的卷积码的可能性。此外，我们还给出了一种在大小为$O(n^{\min\{k,n-k\}})$的有限域上构造速率k/n且轮廓长度L=1的卷积码的显式方法，与相同轮廓长度的已知构造相比实现了更小的域大小。