Abstract Algebra: Theory and Applications · Problem 8.6.17

Isomorphisms — Thomas Judson

Mathematics — Abstract Algebra · hamming-distance-metric

Problem

A metric on a set

X

is a map

d: X \times X \to \mathbb{R}

satisfying the following conditions. ((a))

d(x,y) \geq 0

for all

x, y \in X

; ((b))

d(x,y) = 0

exactly when

x = y

; ((c))

d(x,y) = d(y,x)

; ((d))

d(x,y) \leq d(x,z) + d(z,y)

. In other words, a metric is simply a generalization of the notion of distance. Prove that Hamming distance is a metric on

\mathbb{Z}_2^n

. Decoding a message actually reduces to deciding which is the closest codeword in terms of distance.