On the Deep Holes of Reed-Solomon Codes

Reed-Solomon codes were formally introduced to the world in 1960, and have been both practically and theoretically interesting since their introducion. Used in CD players, long-distance space communication, and computer networks, these simple codes have fascinated coding theorists and theoretical computer scientists for nearly five decades. In addition to being one of the Maximum Distance Seperable codes, Reed-Solomon codes are good for correcting burst errors, and are mathematically simple to describe. They also have interesting connections to a number of problems in computational complexity, which are reviewed in Luca Trevisan’s paper, Some Applications of Coding Theory in Computational Complexity. Here, we seek to understand the structure of Reed-Solomon codes through an examination of their deep holes. Deep holes have been completely characterized for some lattices, and are trivial to characterize in the case of some codes. For Reed-Solomon codes, characterization is neither trivial nor explicitly well-studied, but, both in the original proof that maximum likelihood decoding is NP-Hard 3 and the proof we present here, deep holes seem to be the reason that certain types of decoding are NP-Hard. In pursuit of a characterization, we show a connection between counting the number of points on an absolutely irreducible hypersurface, and determining whether or not a recieved word is a deep hole. We show that the bounded distance decoding problem can be reformulated as the problem of finding a solution to a set of equations. Using an estimation of the number of points on an arbitrary irreducible hypersurface, we show that, for some Reed-Solomon codes, recieved words of small degree cannot be deep holes. We conjecture that the deep holes of Reed-Solomon codes are easy to classify when the evaluation set of the code is a complete field. This conjecture, if true, would suggest that decoding such codes is easy, or that a new proof technique would be required to prove hardness. An additional contribution of this paper is that we do not use the check matrix of the code to prove hardness. Our approach to the problem, while simple, appears to be unique.