Algebraic List-Decoding in Projective Space: Decoding With Multiplicities and Rank-Metric Codes

Abstract

The problem of list decoding algebraic subspace codes and rank-metric codes is considered. We develop two separate methods, via two different approaches, for list decoding subspace codes and rank-metric codes. These methods provide, for certain code parameters, improved tradeoffs between rate and error-correction capability than that of the Koetter-Kschischang codes, in the domain of subspace codes, and than that of the Gabidulin codes, in the domain of rank-metric codes, and several other extensions thereof. In the first approach, we introduce the notion of root multiplicities for a certain sub-ring of the ring of linearized polynomials. In the list-decoding algorithm, multiple roots are enforced for the interpolation polynomial in order to achieve an improved error-correction radius for an extended family of Koetter-Kschischang subspace codes. The normalized error-correction radius for this approach is τ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> =2(L+1)/(r+1)-1-L(L+1)(L+n)/r(r+1)R, where L is the maximum list size, n is the subspace code dimension, R is the rate of the code, and r is the multiplicity parameter. In the second approach, we construct a folded version of Koetter-Kschischang codes. A linear-algebraic list-decoding algorithm is proposed for these codes that achieves the error-correction radius τ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> =s(1-sR), where s is the folding parameter. As opposed to the first approach, the size of output list in the second approach depends on the underlying field size and is at most q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m(s-1)</sup> , where q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> is the size of the field that message symbols are chosen from. It is also shown that the output list size is 1, with high probability, in a probabilistic setting. We utilize the techniques of the second approach in the domain of rank-metric codes to construct folded Gabidulin codes to enable a linear-algebraic list-decoding algorithm for such codes. Our algorithm makes it possible to recover.