Question

Complete the proof of Theorem 16.3 by showing that if a code detects t errors,then the Hamming distance between any two codewords is at least $t+1$.

Short answer

It is proved that if a code detects t errors, then the Hamming distance between any two codewords is at least$t+1$

Step-by-step solution

 Step 1: Definition 

A linear code is said to detect terrors if the received word in anytransmission with at least one, but no more thant errors, is not a codeword.

Proving that if a code detects t errors, then the Hamming distance between any two codewords is at least t + 1

Let’s assume that for this linear code codewords u and v are such that

$d\left(u,v\right)<t+1$

This implies,

$d\left(u,v\right)⩽t$

But we know from the definition of linear code which detects t errors, that any received message word having more than t errors is not a codeword. So, in linear code it is not possible to have two codewords with more than t errors, this implies$d\left(u,v\right)⩽t$ is not possible, therefore our assumption is wrong.

Hence it is proved that if a code detects t errors, then the Hamming distance between any two codewords is at least$t+1$