Finite-Length Analysis of BATS Codes

Abstract

BATS codes were proposed for communication through networks with packet loss. A BATS code consists of an outer code and an inner code. The outer code is a matrix generation of a fountain code, which works with the inner code that comprises random linear coding at the intermediate network nodes. In this paper, the performance of finite-length BATS codes is analyzed with respect to both belief propagation (BP) decoding and inactivation decoding.Our results enable us to evaluate efficiently the finite-length performance in terms of the number of batches used for decoding ranging from 1 to a given maximum number, and provide new insights on the decoding performance.Specifically, for a fixed number of input symbols and a range of the number of batches used for decoding, we obtain recursive formulae to calculate respectively the stopping time distribution of BP decoding and the inactivation probability in inactivation decoding. We also find that both the failure probability of BP decoding and the expected number of inactivations in inactivation decoding can be expressed in a power-sum form where the number of batches appears only as the exponent. This power-sum expression reveals clearly how the decoding failure probability and the expected number of inactivation decrease with the number of batches. When the number of batches used for decoding follows a Poisson distribution, we further derive recursive formulae with potentially lower computational complexity for both decoding algorithms. For the BP decoder that consumes batches one by one, three formulae are provided to characterize the expected number of consumed batches until all the input symbols are decoded.

Intelligent cognitive spectrum collaboration: Convergence of spectrum sensing, spectrum access, and coding technology

“…For example, Ref. [91] proposed a greedy method to find the optimal degree distribution for finite original symbols. Reference [92] used prior information on better degree distribution to greatly reduce the number of optimization variables.…”

Joint BATS Code and Periodic Scheduling in Multihop Wireless Networks

“…The design of the outer code and the inner code can be largely separated given the destination node transfer matrix rank distribution (called the rank distribution henceforth). The design of the outer code has been extensively studied in [8], [18], [19]. In [8], a sufficient condition for the BP decoder to recover a given fraction of the input packets with high probability was obtained.…”

“…It has been verified theoretically for certain special cases and demonstrated numerically for general cases that BATS codes can achieve rates very close to the expected rank of the given rank distribution of the transfer matrices, which is an upper bound on the achievable rate of BATS codes. In [18], the finite-length performance of the outer code has been studied and showed that inactivation decoding can achieve rates very close to the expected rank of the given rank distribution of the transfer matrices when K is relatively small. Based on the performance of the outer code, the main goal of the design of the inner code is to maximize the expected rank of the transfer matrices normalized by certain network communication cost.…”

A Protocol Design Paradigm for Batched Sparse Codes

“…The distribution can be obtained by solving an optimization problem [ 63 ]. When K is small, a better distribution can be obtained via the approach in References [ 65 , 66 , 67 ]. There are also some variants on the degree distribution optimization problem which apply a sliding window [ 68 ] or an expanding window [ 69 ].…”