Can the BMS Algorithm Decode Up to [(dG)-g-1)/2] Errors? Yes, but with Some Additional Remarks

Abstract

It is a well-known fact [7], [9] that the BMS algorithm with majority voting can decode up to half the Feng-Rao designed distance dFR. Since dFR is not smaller than the Goppa designed distance dG, that algorithm can correct up to $\lfloor \frac{d_G-1}{2} floor$ errors. On the other hand, it has been considered to be evident that the original BMS algorithm (without voting) [1], [2] can correct up to $\lfloor \frac{d_G-g-1}{2} floor$ errors similarly to the basic algorithm by Skorobogatov-Vladut. But, is it true? In this short paper, we show that it is true, although we need a few remarks and some additional procedures for determining the Groebner basis of the error locator ideal exactly. In fact, as the basic algorithm gives a set of polynomials whose zero set contains the error locators as a subset, it cannot always give the exact error locators, unless the syndrome equation is solved to find the error values in addition.