We propose an algorithm which is an improved version of the KabatianskyTavernier list decoding algorithm for the second order Reed-Muller code RM(2, m), of length n = 2 m , and we analyse its theoretical and practical complexity. This improvement allows a better theoretical complexity. Moreover, we conjecture another complexity which corresponds to the results of our simulations. This algorithm has the strong property of being deterministic and this fact drives us to consider some applications, like determining some lower bounds concerning the covering radius of the RM(2, m) code. AMS Classifications94B05 · 94B35 · 94B65 · 94A60 IntroductionIntroduced by Elias [1] 50 years ago, the concept of list decoding has been recently revived thanks to Sudan's discovery [2] of efficient list decoding algorithms for Reed-Solomon (RS) codes. Despite that there are similarities between Reed-Solomon and Reed-Muller (RM) codes, no efficient list decoding algorithm for RM codes was known until very recently. This article is a rewritten and completed version of Fourquet R., Tavernier C.: List decoding of second order Reed-Muller codes and its covering radius implications, Workshop on Coding and Cryptography 2007, pp. 147-156. R. Fourquet (B)