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Starting from the definition of a weightable quasi-metric we observe that several functional equations are induced in a natural way. Studying these equations we characterize weightable quasi-metrics and show that they define representable total preorders. We also analyze how to retrieve weightable quasi-metrics from real-valued functions satisfying suitable functional equations.

Decoding errors can be seen from the point of view of the receiver or the transmitter. This naturally leads to different functions for the decoding error probability. We study their behaviour and the relation between these two functions. Though both functions are equally good when used to compare two codes with respect to decoding errors only one of them reflects in general the properties such a function should have. This is not the function one usually considers in the literature when studying decoding errors. Both functions coincide only if the underlying code is perfect. The investigations in this paper can be seen as a continuation of earlier work of MacWilliams (see chap. 16.1 in [2]).

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