A Lower Bound on the Field Size of Convolutional Codes With a Maximum Distance Profile and an Improved Construction
Zitan Chen · IEEE Transactions on Information Theory · 2023
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 ratek/nand a maximum distance profile require a field of size at least exponential innfor general code parameters. At the same time, the only known lower bound on the field size is the trivial bound that is linear inn. In this paper, we show that a finite field of size ΩL(nL-1) is necessary for constructing convolutional codes with ratek/nand a maximum distance profile of lengthL. As a direct consequence, this rules out the possibility of constructing convolutional codes with a maximum distance profile of lengthL≥ 3 over a finite field of sizeO(n). Additionally, we also present an explicit construction of convolutional code with ratek/nand a maximum profile of length L = 1 over a finite field of sizeO(nmin{k,n-k}), achieving a smaller field size than known constructions with the same profile length.