Linearized Polynomial Interpolation and Its Applications

In this paper, we first propose an interpolation algorithm in a well ordered free module of a linearized polynomial ring, and then use this algorithm to decode several important families of codes, Gabidulin codes, Kötter and Kschischang (KK) codes and Mahdavifar and Vardy (MV) codes. Our decoding algorithm for Gabidulin codes is different from the polynomial reconstruction algorithm by Loidreau. When applied to decode KK codes, our interpolation algorithm is equivalent to the Sudan-style list-1 decoding algorithm proposed by Kötter and Kschischang for KK codes. The interpolation approach is also capable of solving the interpolation problem for the list decoding of MV codes proposed by Mahdavifar and Vardy, and has a lower complexity than Gaussian elimination. An interpolator for list decoding of MV codes has also been implemented in hardware and the synthesis results show that it leads to better throughput and efficiency than Gaussian elimination.