|| Solving Monoshift Systems and Applications in Random Coding
||Yiheng Zhang, Ximing Fu, Xuanchen Wu, Shenghao Yang, Kenneth W. Shum, The Chinese University of Hong Kong, Shenzhen, China|
||D3-S5-T2: Topics in Coding II
||Wednesday, 14 July, 23:20 - 23:40
||Wednesday, 14 July, 23:40 - 00:00
A monoshift matrix is a matrix that has binary poly- nomails of degree at most 1 as entries, and a monoshift system is a system of linear equations over polynomials with a monoshift coefficient matrix. We propose an algorithm called augmented elimination to reduce a monoshift matrix to a form called augmented echelon form of degree at most 1. The monoshift system in augmented echelon form can be solved efficiently by successive cancellation. We further derive a recursive formula of the rank distribution of a uniformly random monoshift matrix. For a square uniformly random monoshift matrix, the full- rank probability increases to 1 almost exponentially fast as the matrix size increases. This is quite different compared with square random matrices over a fixed finite field, where the full-rank probability decreases when the matrix size increases. Certain coding problems can benefit from this special property of monoshift systems, as demonstrated by two applications: distributed storage with decentralized encoding and batched network coding.