Designing of Fountain Codes with Short Code-Length

1. Designing of Fountain Codes with Short Code-Length Hongjie Zhu ,Chao Zhang ,Jianhua Lu International Workshop on Signal Design and Its Applications in Communications, 2007. IWSDA 2007. 3rd

2. Outline • Introduction • Expected Ripple Size (ERS) Function • Modified ERS Function with Variance Compensation • Simulation Results

3. Introduction • Transport Control Protocol (TCP) : • Can guarantee the data integrity. • But the feed-back mode has many restrictions. • Fountain codes : • Also add redundant information. • But rateless • Low complexity in both encoding and decoding algorithms. • LT code • High Probability of Complete Decoding (PCD) can be achieved. • Raptor code • Combining a pre-code and a weakened LT code together. • Higher PCD and lower encoding and decoding complexity.

4. Introduction • The code-length practicable is usually on the order of or higher. • In some applications, the delay caused by the buffering process is barely acceptable. • The delay will be tolerable if the code-length can be reduced to 2000 or less. • Fountain codes with short code-length are needed.

5. Introduction • The degree distribution of encodingsymbols plays an important role in improving the decoding performance. • Expected Ripple Size (ERS) function: • Track the variation of the expected ripple size. • Have one-to-one correspondence with the degree distribution. • We can get intuitional explanation to the decoding performance. • Help us to find useful restrictions or heuristic conditions for the designing work of degree distribution. • But the occurring of large deviation turns to be remarkable when it comes to short-length codes. • Using the compensation functions to solve this problem.

6. Expected Ripple Size (ERS) Function • Ripple: At the first step all encoding symbols with one neighbor are released to cover their unique neighbor. The set of covered input symbols that have not yet been processed is called the ripple. • We can define Ripple Size Process (RSP) as a stochastic process • Since the ripple size varies during the decoding process • The ERS function (the expected function of RSP): i: the decoding step r(i): the expected number of the released encoding symbols at decoding step i

7. Expected Ripple Size (ERS) Function • r(i): the expected number of the released encoding symbols at decoding step i • We can rewrite • P: the k x 1 degree distribution vector • T: the k x k release rate matrix • T(i, j): the release rate of symbols of degree j at decoding step i. One-to-one correspondence The problem of finding good degree distributions Full • Designing a fine ERS function

8. Expected Ripple Size (ERS) Function • Find a degree distributions that can build high ERS functions. • The choosing of ERS function becomes an optimizing problem with a series of restricting conditions. • Goal for LT code: • High PCD(Probability of Complete Decoding) • Maintain the ERS function with high constant value • Goal for weakened LT codes in Raptor codes: • High decoding ratio • The early termination of decoding must be minimized. • Failings appear in the last phase of decoding process can be tolerable. • Choose a degressiveERS.

9. Expected Ripple Size (ERS) Function • The probability of decoding failing is proportional to the reciprocal of ripple size: • Model the limitation of the ERS function: Assume Constant factor Ripple size Probability of decoding failing Constant

10. Expected Ripple Size (ERS) Function • Goal: • Maximize the decoding ratio • Minimize the number of undecoded symbols. • By applying the Lagrange multiplier method

11. Modified ERS Function with Variance Compensation • Goal for LT code: • High PCD(Probability of Complete Decoding) • Maintain the ERS function with high constant value. • The random fluctuation of ripple size must be kept little enough. • But the deviation of the real size of ripple from its average value is not neglectable.

12. Modified ERS Function with Variance Compensation • The extent of deviation varies throughout the decoding process

13. Modified ERS Function with Variance Compensation

14. Modified ERS Function with Variance Compensation • The lower 3σ bound is suitable to be the base line of designing. • A modified ERS function • Can be generated by adding a proper multiple of the standard deviation function to certain desired baseline function. • The baseline becomes a statistical low bound of the RSP.

15. Modified ERS Function with Variance Compensation

16. Modified ERS Function with Variance Compensation • How to get the variance function? • Many codes with different parameters are examined, and similar figures are gained. • Numerical computation of the variance function is adequate. • Sophisticated theoretical analysis on the decoding process is avoided.

17. Simulation Results • Take the modified ERS function as the key restriction of optimization. • This method employs linear programming to find a solution of the degree distribution.

18. Simulation Results • Different modified ERS functions can be made. • By using different baselines for various optimization purposes. • LT code: • Minimizing the probability of decoding failing. • wLTcode: • Maximizing the decoding ratio.

19. Simulation Results

20. Simulation Results