Non-volatile memories (NVMs), such as Magnetic RAM and Resistive RAM, have been considered as the potential working or storage memories in the next generation computer architectures, thanks to the various merits, such as non-volatility, low power and high speed etc. However, new technology brings simultaneously new challenges, e.g. reliability issue, before practical applications. Compared with conventional memories, the errors in NVMs are generally asymmetric, resulting in different failure rates for 0-1 and 1-0 bit flipping. Error correcting codes (ECCs) are common solutions for protecting memories from errors. The most widely used ECCs are the single error correction and double error detection (SEC-DED) codes. Unfortunately, they are primarily designed for correcting symmetric errors and their error correction capabilities are limited to only one bit. Regarding the failure characteristics (e.g., multi-bit and asymmetric) of NVMs, conventional SEC-DED codes are not efficiently applicable. This paper proposes an extended coset decoding scheme for NVMs. Our simulation results with a Hamming code (with Hamming distance of only three) as an example show the effectiveness of the proposed decoding scheme. The proposed decoding scheme can also be extended to other linear block codes and is rather suitable for scenarios with multi-bit asymmetric error features.