Soliton-Like LT Codes over GF(4) and GF(8)

Abstract

The Luby Transform (LT) codes are the first practical realization of fountain codes. LT codes were designed for efficient use over erasure channels. However, the degree distributions designed for the Binary Erasure Channel (BEC), such as the Robust Soliton Distribution (RSD) which are asymptotically and universally optimal over the BEC, are not optimal over noisy channels. Research has been done to improve performance over noisy channels and in finite length codes by changing the degree distribution or information sampling method. Many distributions that performed better than the RSD still shared some of its properties. These distributions fit under the umbrella of soliton-like distributions. In this work, we consider using non-binary LT codes to improve the Bit Error Rate (BER) and realized rate of the system with an acceptable increase in complexity. While testing different degree distributions on 4-ary and 8-ary LT code, the RSD was not performing well. This led to the design of the fundamental degree distribution (FDD), which falls under the category of soliton-like distributions. The parameters for this distribution are found through Monte Carlo numerical optimization. This new degree distribution was able to show improvements when moving to higher order fields in terms of BER and realized rate. It is clear that there is added complexity when moving to higher order fields. Some of this complexity is removed in this work by performing no multiplications over the field at intermediate nodes, as it was shown to provide negligible gains. As well, we look at shaping the left degree distribution (LDD) to further increase the improvements in the non-binary applications. We show that by strategically connecting the encoded symbols to the input symbols with the lowest degree, we can see improvements in the BER and realized rate of 4-ary LT codes. The realized rate increases from around 0.75 to 0.85 at an SNR per bit of 16 dB, simply by shaping the LDD for our non-systematic short length codes, with k = 100 at a negligible added complexity.