Geodesic
Natural metrics and their properties. P.~1. Submetrics and overmetrics
Matematičeskie voprosy kriptografii, Tome 2 (2011), pp. 49-74

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Criteria for integer-valued function $\mu\colon X\times X\to\{0,1,\dots\}$ to be a metric (where $X$ is a finite set) are given. Notions of submetric, overmetric, natural and canonical metrics are introduced. Classification of metrics admitting no more than 5 values is constructed, some of their submetrics and overmetrics are described. 
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     author = {B. A. Pogorelov and M. A. Pudovkina},
     title = {Natural metrics and their properties. {P.~1.} {Submetrics} and overmetrics},
     journal = {Matemati\v{c}eskie voprosy kriptografii},
     pages = {49--74},
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     volume = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MVK_2011_2_a3/}
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B. A. Pogorelov; M. A. Pudovkina. Natural metrics and their properties. P.~1. Submetrics and overmetrics. Matematičeskie voprosy kriptografii, Tome 2 (2011), pp. 49-74. http://geodesic.mathdoc.fr/item/MVK_2011_2_a3/