Some Results on List Decoding of Interleaved Reed-Solomon Codes with the Extended Euclidean Algorithm Sabine Kampf, Martin Bossert Institute of Telcommunications and Applied Information Theory Ulm University, Ulm, Germany {sabine.kampf, martin.bossert}@uni-ulm.de Sergey Bezzateev Saint Petersburg State University of Aerospace Instrumentation St. Petersburg, Russia bsv@aanet.ru Abstract—A recently presented algorithm for joint de- coding of interleaved Reed-Solomon (IRS) codes based on the extended Euclidean algorithm is recalled and it will be shown that with a simple extension a list decoding algorithm is obtained. It is also shown how decoding works in case when the error weight is not the same for all codewords. I. I NTRODUCTION Collaborative or joint decoding of interleaved Reed- Solomon codes has been introduced in [2] and [3] and has been in the focus of research during the last years. The idea is to use several parallel codewords such that the error locator is the same for all codes while the syndroms give independent information (at least with high probability). Several decoding algorithms have been developed: In [4] a multi-sequence shift register algo- rithm was used to correct errors up to the maximum correcting radius. However the algorithm can correct a unique error or fails to correct. In [5], a new method for decoding interleaved RS codes has been introduced that is based on the Euclidean algorithm. After recalling the important properties of the ex- tended Euclidean algorithm and the basic decoding al- gorithm, this paper will show how the algorithm can be extended to work as a list decoding algorithm. Then we show that the algorithm is insusceptible to errors with different weights and how this fact can be used avoid having to determine the number of errors before starting the decoding process. II. NOTATIONS AND DEFINITIONS It is well known, that the Euclidean algorithm calcu- lates the greatest common divisor of two polynomials A(x) and B(x) with coefficients from GF(q). Theorem 1 [1] For any two polynomials A −1 (x)= B(x),A 0 (x)= A(x) with deg A −1 (x) ≥ deg A 0 (x) the greatest common divisor D(x) is calculated by the following recursion: A −1 (x) = A 0 (x)Q 1 (x)+ A 1 (x) A 0 (x) = A 1 (x)Q 2 (x)+ A 2 (x) . . . A j −2 (x) = A j −1 (x)Q j (x)+ A j (x) A j −1 (x) = A j (x)Q j +1 (x),A j +1 (x)=0. (1) where deg A i−1 > deg A i and the last nonzero remain- der A j (x) is the greatest common divisor D(x) of two polynomials A(x) and B(x) D(x)= GCD(A(x),B(x)) = A j (x) (2) Any remainder can then be written as a combination of the original polynomials A(x) and B(x), this is usually called the extended Euclidean algorithm. Theorem 2 [1] Let A(x) and B(x) be polynomi- als with deg A(x) ≤ deg B(x) and let D(x) = GCD(A(x),B(x)). Then there exist polynomials U i (x) and V i (x) such that U i (x)A(x)+ V i (x)B(x)= A i (x), i =1, ..., j (3) where A j (x) = D(x), deg U i (x) < deg A(x), deg V i (x) < deg A(x), and the polynomials A i (x) are calculated by the Euclidean algorithm from Equation 1. (3) clearly shows, that with the extended Euclidean algorithm we are able to calculate polynomials A i (x) and U i (x) with the property A i (x)= A(x) · U i (x) mod B(x), (4)