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Justesen has shown that concatenating a class of binary codes with a Reed-Solomon (RS) code produces asymptotically good codes. For low rates, the value of the ratio of minimum distance to code length for such codes is substantially lower than that known to be achievable by the Zyablov bound. In this paper, we present a small class of binary codes with some useful properties. This class is then used in Justesen's construction to produce codes that have relatively large values of for low rates.