Comments and Corrections

Abstract

We would like to make some comments on the flowchart of the Chase-based soft-decision decoding algorithm, i.e., Algorithm 2, in our previous paper [1]. We note that there is a mistake in Fig. 1 in [1]; that is, v1 is initialized to z where z denotes the hard-decision vector of the channel observation r. In our previous simulations, we initially set λ(r,v1) = λ(r,z) = +∞ which guarantees that v2 rather than z is chosen where v2 represents the first decoded codeword. Then v2 will be compared with other valid codewords obtained from different test patterns. Nevertheless, perhaps the first valid codeword v2 and the initial v1 (namely z) satisfy condition (13) in [1] and thus terminate the decoding process. In this case, the codeword v(v = v2) may not be the optimal candidate of the soft-decision decoder since Theorem 1 in [1] is correct if and only if v1 and v2 are both valid codewords and meet condition (13). Fortunately, the performance curves of the (89, 45, 17) QR code presented in [1] were accurate despite the mistake mentioned above. In reality, the true error-rate performance can also be obtained for the (73, 37, 13) and (71, 36, 11) QR codes by utilizing the flowchart of Algorithm 2 in [1]. However, we observed a significant performance gap as illustrated in Fig. 1 when the (23, 12, 7) QR code, i.e., the Golay code, was considered. We guess that such a loss results from the Golay code’s ‘perfect code’ property. It is worth to note that the decoding flowchart in [1] is not optimal from decoding speed point of view since the second valid codeword still needs to be solved even if the first one is the maximum-likelihood (ML) codeword. Actually, it is often the case at high signal-to-noise ratio (SNR). To overcome this, a new flowchart which integrates two different sufficient optimality conditions is developed in the next section.