List Error-Correction with Optimal Information Rate (Invited Talk)

Abstract

The construction of error-correcting codes that achieve the best possible trade-off between information rate and the amount of errors that can be corrected has been a long sought-after goal. This talk will survey some of the work on list error-correction algorithms for algebraic codes {[}8,5], culminating in the construction of codes with the optimal information rate for any desired error-correction radius {[}7,4]. Specifically, these codes can correct a fraction p of worst-case errors (for any desired 0 < p < 1) with rate 1 - p - is an element of for any constant is an element of > 0. We will describe these codes, which are called folded Reed-Solomon codes, and give a peek into the algebraic ideas underlying their list decoding. Over the years, list-decodable codes have also found applications extraneous to coding theory {[}3, Chap. 12], including several elegant ones in cryptography. The problem of decoding Reed-Solomon codes (also known as polynomial reconstruction) and its variants from a large number of errors has been suggested as an intractability assumption to base the security of protocols on {[}6]. Progress on list decoding algorithms for algebraic codes has led to cryptanalysis of some of these schemes. It is interesting to note that the line of research that eventually led to the above-mentioned result for folded Reed-Solomon codes can be traced back to a cryptographic assumption concerning simultaneous polynomial reconstruction and algorithms for decoding ``interleaved{''} Reed-Solomon codes that it inspired {[}1,2]. Given the cryptographic theme of the WITS conference, we will also briefly allude to the above connection in the talk.