Unified Upper Bounds on the ML decoding Error Probability of Spinal Codes over Fading Channels

TL;DR

This paper develops tight upper bounds for the block error rate of Spinal codes over various fading channels, providing a comprehensive framework for performance evaluation in finite block length regimes.

Contribution

It introduces new analytical bounds for Spinal codes' error probability over fading channels, including a novel bound without Gallager's method, and demonstrates their tightness through simulations.

Findings

Derived explicit BLER upper bounds for Rayleigh, Nakagami-m, and Rician channels.

Proved the tail transmission pattern (TTP) remains optimal for ML decoding.

Validated bounds' tightness via simulations.

Abstract

Performance evaluation of particular channel coding has been a significant topic in coding theory, often involving the use of bounding techniques. This paper focuses on the new family of capacity-achieving codes, Spinal codes, to provide a comprehensive analysis framework to tightly upper bound the block error rate (BLER) of Spinal codes in the finite block length (FBL) regime. First, we resort to a variant of the Gallager random coding bound to upper bound the BLER of Spinal codes over the fading channel. Then, this paper derives a new bound without resorting to the use of Gallager random coding bound, achieving provable tightness over the wide range of signal-to-noise ratios (SNR). The derived BLER upper bounds in this paper are generalized, facilitating the performance evaluations of Spinal codes over different types of fast fading channels. Over the Rayleigh, Nakagami-m, and Rician fading channels, this paper explicitly derived the BLER upper bounds on Spinal codes as case studies. Based on the bounds, we theoretically reveal that the tail transmission pattern (TTP) for ML-decoded Spinal codes remains optimal in terms of reliability performance. Simulations verify the tightness of the bounds and the insights obtained.