# Robust Soliton Distribution

## What is the Robust Soliton Distribution?

The Robust Soliton distribution is a way to construct a good degree of encoding symbols [1]. LT (Luby transform) code analysis is based on analysis a decoding algorithm called the LT processes. Each LT code symbol in the process is generated by randomly selecting the degree of the code symbol, d, between 1 and k. The distribution is constructed so that the decoder can recover data from more than k code symbols with probability > 1 – δ, where 0 < δ < 1 [2].

Robust soliton distributions are used in a variety of applications where data loss is a possibility. Some of the most common applications include:

• Data transmission: Robust soliton distributions can be used to protect data during transmission over unreliable channels, such as wireless networks or satellite links.
• Data storage: Robust soliton distributions can be used to protect data stored on unreliable media, such as flash drives or hard drives.
• Data backup: Robust soliton distributions can be used to protect data during backup and restore operations.
• Data compression: Robust soliton distributions can be used to compress data without losing any information. This can be useful for applications where data needs to be stored or transmitted efficiently.

Robust soliton distributions are a relatively new technology capable of safeguarding data in scenarios where data loss is possible. Their applications include data transmission, storage, backup, and compression. Specifically, they can protect data over unreliable channels such as wireless networks or satellite links, unreliable media such as flash drives or hard drives, and during backup and restore operations. Additionally, robust soliton distributions can compress data without losing any information, which is useful for efficient storage or transmission. The efficacy of robust soliton distributions in a variety of applications has been well-established. As the demand for reliable data protection continues to surge, the usage of robust soliton distributions is projected to broaden as well.

## Ideal vs. robust soliton distribution

The ideal soliton distribution is a probability distribution with a single parameter K. Its probability mass function (PMF) is [3]:

The ideal soliton distribution has a peak at 2, which means that a majority of code symbols have a degree of 2. This makes the code vulnerable to erasures, since a code symbol’s erasure with a degree 2 will lead to the loss of the entire ripple. Ripples are sequences of symbols used to cover two input symbols. Each ripple starts with one symbol and gradually increases in size to cover more input symbols. This process continues until all input symbols are covered. By ensuring the constant size of the ripple, the ideal soliton distribution guarantees that the decoder will always have enough symbols to cover all the input symbols. The ideal soliton distribution can be an efficient way to encode data. However, it can be sensitive to errors and if too many occur, the ripple may not be able to cover all input symbols, which means that the decoder will fail.

To be less vulnerable to erasures, one can use the robust soliton distribution, which adds a second peak to the distribution at a value of K/R, where K is the number of code symbols and R is the number of information symbols [4].

Since the addition of a peak in the distribution results in greater resilience to erasures, an erasure of a code symbol with degree K/R will only cause the ripple to be lost if the erasure occurs at the first symbol in the ripple. Even with its complexity, the robust soliton distribution is superior to the ideal soliton distribution in resisting erasures. Hence, it is recommended for wireless communication and other applications where erasures are likely to occur.

## References

[1] Project 5: Capstone Project: LT Codes. Retrieved July 14, 2021 from: http://cs.brown.edu/courses/csci1680/f17/content/lt.pdf

[2] Luby, M. (2002). LT Codes, Proc. of the 43rd Annual IEEE Symp. on Foundations of Comp. Sc., pp 271-280, Vancouver, Canada, November.

[3] Tirronen, Tuomas (2005). “Optimal Degree Distributions for LT Codes in Small Cases”. Helsinki University of Technology. CiteSeerX 10.1.1.140.8104.

[4] Joshi et al. Fountain Codes. Retrieved July 14, 2021 from: https://www.andrew.cmu.edu/user/gaurij/FountainCodes.pdf

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