  Spinal codes are known to be capacity achieving over both the additive white
Gaussian noise (AWGN) channel and the binary symmetric channel (BSC). Over
wireless channels, Spinal encoding can also be regarded as an
adaptive-coded-modulation (ACM) technique due to its rateless property, which
fits it with mobile communications. Due to lack of tight analysis on error
probability of Spinal codes, optimization of transmission scheme using Spinal
codes has not been fully explored. In this work, we firstly derive new tight
upper bounds of the frame error rate (FER) of Spinal codes for both the AWGN
channel and the BSC in the finite block-length (FBL) regime. Based on the
derived upper bounds, we then design the optimal transmission scheme.
Specifically, we formulate a rate maximization problem as a nonlinear integer
programming problem, and solve it by an iterative algorithm for its dual
problem. As the optimal solution exhibits an incremental-tail-transmission
pattern, we propose an improved transmission scheme for Spinal codes. Moreover,
we develop a bubble decoding with memory (BD-M) algorithm to reduce the
decoding time complexity without loss of rate performance. The improved
transmission scheme at the transmitter and the BD-M algorithm at the receiver
jointly constitute an "encoding-decoding" system of Spinal codes. Simulation
results demonstrate that it can improve both the rate performance and the
decoding throughput of Spinal codes.
