Question

Find a parity-check matrix for the $\left(15,11\right)$Hamming code.

Short answer

The parity-check matrix for the hamming code (15,11) is:

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

Consider the definition below:

A linear block code's parity check matrix is as follows:

If the generator matrix of a linear block code is in the following form, it is in systematic form.

$G=\left(i\right)A$……………….. (1)

Where I_k denotes the $k\times k$identity matrix. The first k binary symbols in a codeword in a systematic code are the information bits, and the remaining $\left(n-k\right)$ binary symbols are the parity-check symbols.

The parity-check matrix of a code is any $\left(n-k\right)\times n$binary matrix H such that for all codeword c,

$cH\text{'}=0$ …………………. (2)

Obviously;

$G{H}^{\text{'}}=0$ …………………. (3)

Also if G is in systematic form, then

$H=\left[\frac{A}{I_{n-k}}\right]$ …………………. (4)

Hamming code:

Hamming codes are $\left(2^m-1,2^m-m-1\right)$. The parity-check matrix, which is an $m\times \left(2^m-1\right)$matrix, has all binary sequence of length m, except the all-0 sequence, as its columns. The focus is on parity-check matrix for the hamming code $\left(15,11\right)$.

Generator matrix from hamming code

Consider the following (15,11) hamming code:

And thus, the generator matrix is:

Compare with the form in equation (1),

Here,

k=11

n-k=4

Use equation (4), to get;

$H=\left[\frac{A}{I_{n-k}}\right]$

There may be several solutions obtained (including above) which depends on the selection of Hamming code.