Question

If every element of weight $\mathbf{\le }\mathbf{t}$is a coset leader in a standard array for a code C, show that C corrects t errors.

Short answer

If every element of weight $\le t$ is a coset leader in a standard array of a code C, then C corrects t errors.

Step-by-step solution

Conceptual Introduction

Coding theory is the study of a code's characteristics and how well-suited it is to various applications. Data compression, cryptography, error detection, and correction, as well as data transport and storage, all depend on codes.

Determine the introduction

Consider the statement as follows;

If every element of weight $\le t$is a coset leader in a standard array of a code C, then the purpose is to show that C corrects t errors.

Determine the nearest code word to w

If we have: $\mathrm{w}\in \mathrm{B}\left(\mathrm{n}\right)$, then:

$\begin{array}{rcl}w& =& e+v\\ & \in & e+C\\ & & \\ & & \end{array}$

Here, e is the coset leader and v is the code word at the top of the column containing w standard array decoding decodes w as v.

Here, v is the nearest code word to w.

Determine the result

If $u\in c$is any code word, then w-u is an element of w+C.

But w+C is the coset of e (because $e=w-v\in w+C$).

By construction, the coset leader e has smallest weight in its coset.

So, $Wt\left(w-u\right)\ge Wt\left(e\right)$.

Hence, it can be concluded that if every element of weight $\le t$is a coset leader in a standard array of a code C, then C corrects t errors.