Variable-Rate Linear Network Error Correction MDS Codes

TL;DR

This paper introduces variable-rate linear network error correction MDS codes that share the same local encoding kernel across different rates, enabling efficient transmission and error correction in network communications.

Contribution

It proposes a novel concept of variable-rate network MDS codes with shared encoding kernels and provides an efficient construction algorithm and probabilistic analysis.

Findings

The proposed codes reduce storage and transmission resources.

The algorithm's performance is analyzed in terms of field size and complexity.

A lower bound on success probability for random code construction is established.

Abstract

In network communication, the source often transmits messages at several different information rates within a session. How to deal with information transmission and network error correction simultaneously under different rates is introduced in this paper as a variable-rate network error correction problem. Apparently, linear network error correction MDS codes are expected to be used for these different rates. For this purpose, designing a linear network error correction MDS code based on the existing results for each information rate is an efficient solution. In order to solve the problem more efficiently, we present the concept of variable-rate linear network error correction MDS codes, that is, these linear network error correction MDS codes of different rates have the same local encoding kernel at each internal node. Further, we propose an approach to construct such a family of variable-rate network MDS codes and give an algorithm for efficient implementation. This approach saves the storage space for each internal node, and resources and time for the transmission on networks. Moreover, the performance of our proposed algorithm is analyzed, including the field size, the time complexity, the encoding complexity at the source node, and the decoding methods. Finally, a random method is introduced for constructing variable-rate network MDS codes and we obtain a lower bound on the success probability of this random method, which shows that this probability will approach to one as the base field size goes to infinity.