A new method for erasure decoding of convolutional codes

TL;DR

This work introduces a generator-matrix-based erasure decoding method for convolutional codes, applicable even to catastrophic codes, and compares it with the traditional parity-check approach. It develops a formal link between column distances $d^c_j$ and the solvability of linear systems derived from the generator matrix, and defines complete $j$-MDP codes via the generator-matrix viewpoint, including a constructive existence result over large finite fields using powers of a primitive element. The analysis shows that for the regime $k<n-k$, the generator-matrix method offers better erasure-correction performance and guard-space efficiency, with explicit complexity comparisons ($O(((j+1)n-e)^{0.8}((j+1)k)^2)$ vs $O(((j+1)(n-k))^{0.8}e^2)$) and guard-space considerations. Overall, the paper broadens the toolkit for erasure decoding in streaming over erasure channels by leveraging generator matrices and expands the class of optimal codes via complete $j$-MDP constructions, with potential impact on reliable, low-latency communication.

Abstract

In this paper, we propose a new erasure decoding algorithm for convolutional codes using the generator matrix. This implies that our decoding method also applies to catastrophic convolutional codes in opposite to the classic approach using the parity-check matrix. We compare the performance of both decoding algorithms. Moreover, we enlarge the family of optimal convolutional codes (complete-MDP) based on the generator matrix.

Theorems & Definitions (48)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Definition 3
• Proposition 1
• Theorem 3
• proof
• Theorem 4
• Theorem 5