The Existence of Strongly-MDS Convolutional Codes

TL;DR

This paper proves that strongly maximum distance separable convolutional codes exist over sufficiently large finite fields for all parameters, confirming a long-standing conjecture using linear systems theory.

Contribution

It establishes the existence of strongly MDS convolutional codes for all parameters over large finite fields, resolving a major conjecture in coding theory.

Findings

Strongly MDS convolutional codes exist for all parameters.

Existence is proven over large enough finite fields of any characteristic.

The proof employs linear systems theory methods.

Abstract

It is known that maximum distance separable and maximum distance profile convolutional codes exist over large enough finite fields of any characteristic for all parameters $(n,k,\delta)$. It has been conjectured that the same is true for convolutional codes that are strongly maximum distance separable. Using methods from linear systems theory, we resolve this conjecture by showing that, over a large enough finite field of any characteristic, codes which are simultaneously maximum distance profile and strongly maximum distance separable exist for all parameters $(n,k,\delta)$.