The idea of distance between vectors has found many nice applications in both mathematics and engineering. Coding theory (the art of reliable data storage) is fortunate enough to have many distances at hand, such as Hamming, Lee, Euclidean, and so on, which are quite useful for practical and theoretical purposes. In 1995, Brualdi et al. introduced a new poset metric, “determined by a partial order over a finite set, a poset” [1]. This generalizes the Hamming metric and the Niederreiter-Rosenbloom-Tsfasman (NRT) metric. In this book, Firer et al. study the basic questions of coding theory (in light of the classical Hamming metric) with respect to the new poset metric.

Chapter 1 reviews the basic concepts of coding theory (such as duality, syndrome decoding, generalized weight, and certain metric invariants such as packing and covering radius). Chapter 2 continues with partially ordered sets and presents two families of posets, hierarchical posets and multi-chain posets. Chapter 3 focuses on hierarchical posets and obtains results similar to the Hamming metric, such as packing radius, covering radius, perfect codes, and syndrome decoding. The NRT metric has a history of parallel development; one development “corresponds to the P-metric, where P is the union of r chains of length s.” Several coding-theoretic properties do not hold in the multi-chain case, for example, “the packing radius ... is not determined by the minimum distance” and “MacWilliams identities do not hold.”

Chapter 4 answers some of the related questions. Chapter 5 explores several duality theorems such as Wei’s results for generalized weights (using matroids) and MacWilliams-type identities (using group characters). Chapter 6 discusses “coding with general poset metrics, with no restrictions on the poset.” Finally, chapter 7 “present[s] different generalizations and variations of the poset metrics.”

This small but beautiful book presents many new research directions for coding theory students and researchers. Each chapter includes exercises, so it can also be used as a textbook for advanced undergraduate or graduate courses in mathematics and computer science.