The automorphisms and error orbits of Reed – Solomon codes

Computing Error Distance of Reed-Solomon Codes

Under polynomial time reduction, the maximun likelihood decoding of a linear code is equivalent to computing the error distance of a received word. It is known that the decoding complexity of standard Reed-Solomon codes at certain radius is at least as hard as the discrete logarithm problem over certain large finite fields. This implies that computing the error distance is hard for standard Ree...

Decoding of Reed Solomon Codes beyond the Error-Correction Bound

We present a randomized algorithm which takes as input n distinct points {(xi, yi)}i=1 from F × F (where F is a field) and integer parameters t and d and returns a list of all univariate polynomials f over F in the variable x of degree at most d which agree with the given set of points in at least t places (i.e., yi = f(xi) for at least t values of i), provided t = Ω( √ nd). The running time is...

Decoding of Reed Solomon codes beyond the error-correction bound

Quantum Reed-Solomon Codes

During the last years it has been shown that computers taking advantage of quantum mechanical phenomena outperform currently used computers. The striking examples are integer factoring in polynomial time (see [8]) and finding pre– images of an n–ary Boolean function (“searching”) in time O( √ 2n) (see [5]). Quantum computers are not only of theoretical nature—there are several suggestions how t...

Reed-Solomon Error-correcting Codes The Deep Hole Problem

In many types of modern communication, a message is transmitted over a noisy medium. This creates a chance that the message will become corrupted. The purpose of an error-correcting code is to add some redundant information to the message which allows the receiver to detect and correct those errors accrued during the transmission. We will study the famous Reed-Solomon code (found in QR codes, c...