Digital communication, embodied in such applications as cell phones and wireless Internet, by now pervades our daily lives, while digital storage devices, such as CDs, DVDs, and computer disk drives, have become the principal means of preserving our information. In the "information age" in which we all now live, the need for reliable transmission and storage of digital data is of paramount importance. What makes such reliable transmission and storage possibles are error-correcting codes, first conceived by Claude Shannon over 50 years ago. Indeed, as you are reading these lines, millions of error-correcting codes are decoded every minute, using efficient algorithms implemented in custom VLSI circuits. At least 75% of these circuits decode Reed-Solomon codes, invented by Irving Reed and Gustave Solomon in the 1960s. In the four decades since their invention, Reed-Solomon codes have been extensively studied and ingenious decoding algorithms for these codes have been developed. What has been realized only recently, however, is that Reed-Solomon codes can correct many more errors than previously thought possible! In a series of theoretical breakthroughs, Sudan, Guruswami-Sudan, and Koetter-Vardy have made state-of-the-art Reed-Solomon decoders out of date. At least in principle, we can now achieve much better performance with the same codes. The goal of this project is to follow-up on this exciting recent work and to follow this line of research through to its ultimate potential, in theory as well as in practice.
In order to attain this goal, we plan a broad line of attack. On one hand, the proposed investigation will address deep theoretical questions. Can one exceed the Guruswami-Sudan decoding radius? What is the optimal multiplicity assignment for algebraic soft-decision decoding? How can iterative decoding methods be applied to Reed-Solomon codes? On the other hand, we intend to go all the way to the first-ever VLSI implementation of a soft-decision Reed-Solomon decoder. The proposed VLSI architecture aims for high speed and low power dissipation. Thus complexity considerations, inherently motivated by the practice of VLSI design, will be paramount throughout our investigation. Specifically, The main topics to be investigated are: (1) Multivariate interpolation decoding beyond the Guruswami-Sudan radius; (2) Probabilistic model and multiplicity assignment schemes for algebraic soft-decision decoding; (3) Iterative methods for soft decision decoding of Reed-Solomon codes; (4) Analytic bounds on the performance of maximum-likelihood and suboptimal decoders; and (5) Complete VLSI implementation of a state-of-the-art soft-decision Reed-Solomon decoder in an FPGA and/or ASIC.
