Question

Suppose a codeword u is transmitted and w is received. Show that standard-array decoding will decode w as u if and only if$w-u$ is a coset leader.

Short answer

It has been shown that standard-array decoding will decode w as u if and only if$w-u$ is a coset leader.

Step-by-step solution

Result used

Coset leader is an element of smallest weight in the coset and listed first in the row.

Decoding rule is: Decode a received word w as the codeword at the top of the column in which w appears.

Given that

Given that acodeword u is transmitted and w is received.

Let e be the coset leader.

Proof

According to the decoding rule, w will be decoded as codeword which is in the top of the column in which w appears. If w appears in the row with coset leader e, then as per the construction of array $w=e+u\text{'}$.

This implies $u\text{'}=w-e$

This further implies that w will be decoded as u if and only if $w-e=u$.

This happens only when$w-u=e$ is a coset leader.

Hence, standard-array decoding will decode w as u if and only if$w-u$ is a coset leader.