Question

Let${\mathbf{e}}_{\mathbf{i}}\mathbf{=}\mathbf{00}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{010}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{EB}\left(n\right)$ have 1in coordinate iand0elsewhere. IfHis a matrix with n rows, show that e_iH is the ithrow of H.

Short answer

It is shown as: wH=e_iH.

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.

Introduction

Theorem:Consider C as an (n,k) code, which has standard generator matrix G and parity-check matrix H. If and only if wH=0, an element w in B(n) becomes a code word.

Consider the following statement:

Let ${\mathrm{e}}_{\mathrm{i}}=00...010...00\in \mathrm{B}\left(\mathrm{n}\right)$have 1 in coordinate i and 0 elsewhere.

If H is a matrix with n rows, then the purpose is to show that e_iH is i-th row of H.

Determine the proof

As a code word u is transmitted with exactly one error in coordinate i and received as W, then w-u=e_i.

By theorem state above,

$\begin{array}{rcl}wH& =& \left(e_i+u\right)H\\ & =& \left(e_{i}H+uH\right)\\ & =& \left(eH+0\right)\end{array}$

Or,

wH=e_iH.

Therefore, it is proved.