Power Decoding of Reed-Solomon Codes Revisited

Power decoding was originally developed by Schmidt, Sidorenko and Bossert for low-rate Reed–Solomon codes (RS) [1], and is usually capable of decoding almost as many errors as the Sudan decoder [2] though it is a unique decoder. When an answer is found, this is always the closest codeword, but in some cases the method will fail; in particular, this happens if two codewords are equally close to the received. With random errors, this happens exceedingly rarely, though. The decoder builds on the surprising fact that a received word coming from a low-rate RS code can be “powered” to give received words of higher-rate RS codes having the same error positions. For each of these received words, one constructs a classical key equation by calculating the corresponding syndromes, and solving these simultaneously for the same error locator polynomial constitutes the decoder. Gao gave a variant of unique decoding up to half the minimum distance [3]: in essence, his algorithm uses a different key equation and with this finds the information polynomial directly. We here show how to easily derive a variant of Power decoding for Generalised RS (GRS) codes, Power Gao, where we obtain multiple of Gao’s type of key equation, and we solve these simultaneously. Our exposition does not need familiarity with the original Power decoding, “Power syndromes”. We show that our new method is equivalent to the Power syndromes decoding in the sense that they either both fail or both succeed for a given received word. However, some advantages of Power Gao which can be mentioned are: • The information is obtaind directly and one does not need root finding in the error locator and Forney’s formula. • It supports 0 as evaluation point of the GRS code, which Power syndromes does not. • Some properties seem easier to analyse; in particular, we show that whether Power Gao fails or not depends only on the error and not on the sent codeword. This was not known before for Power syndromes. The main drawback is that the size of the polynomials involved in the decoding are slightly larger: approximately of degree n, where Power syndromes uses approximately degree n − k. However, we are dealing with low-rate GRS codes, so this difference is small. We briefly sketched Power Gao already in [4], but its behaviour was not well analysed and its relation to Power syndromes not examined. In Section II we derive the powered Gao key equations, and in Section III we will briefly describe how to solve them simultaneously. In Section IV we derive the decoding radius, and in Section V the other mentioned properties.