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Cumulative weight enumerators of random linear codes are introduced, theirasymptotic properties are studied, and very sharp thresholds are exhibited; asa consequence, it is shown that the asymptotic Gilbert-Varshamov bound is avery sharp threshold point for the density of the linear codes whose relativedistance is greater than a given positive number. For arbitrary random codes,similar settings and results are exhibited; in particular, the very sharpthreshold point for the density of the codes whose relative distance is greaterthan a given positive number is located at half the asymptoticGilbert-Varshamov bound.

Cumulative Distance Enumerators of Random Codes and their Thresholds