Question

Let C be the $\left(4,2\right)$code with standard generator matrix $\mathbf{G}\mathbf{=}\left(\begin{array}{cccc}1& 0& 1& 1\\ 0& 1& 0& 1\end{array}\right)$. Construct a standard array for C and find the syndrome of each coset leader.

Short answer

The set of coset leaders is $\left\{0000,0101,1010,1111\right\}$, and the corresponding syndromes are as $\left\{00,11,01,10\right\}$.

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

Let C be the $\left(4,2\right)$code

$\left\{0000,0101,1010,1111\right\}$

The generator matrix is;

$\mathrm{G}=\left(\begin{array}{cccc}1& 0& 1& 1\\ 0& 1& 0& 1\end{array}\right)$

The focus is to create the standard array for C and identify each coset leader's syndrome.

Forming table for e1+C 

Now, the given generator matrix is;

$G=\left(\begin{array}{cccc}1& 0& 1& 1\\ 0& 1& 0& 1\end{array}\right)$

The corresponding parity-check matrix is;

$H=\left(\begin{array}{cc}1& 0\\ 0& 1\\ 1& 1\\ 0& 1\end{array}\right)$

From the elements of B(4) not in C, choose one of the smallest weight (which in this case is weight 1), say $e_1=1000$. From its coset $e_1+C$ by adding e₁ successively to the elements of C and list the coset elements, with data-custom-editor="chemistry" ${\mathrm{e}}_1+\mathrm{C}$directly below for each data-custom-editor="chemistry" $\mathrm{c}\in \mathrm{C}$.

C	0000	0101 1010 1111
e1+C	1000	1101 0010 0111

Table 1

Creating for e2=1000

Similarly, for $e_2=0100$

C	0000	0101 1010 1111
e2+C	0100	0001 1110 1011

Table 2

Completing the table with all elements of B(4)

Repeat this until every elements of is on the table. This table is complete.

C	0000	0101 1010 1111
e1+C e2+C e3+C	1000 0100 1100	1101 0010 0111 0001 1110 1011 1001 0110 0011

Table 3

Finding corresponding syndrome

Here, the top row indicated by C is the set of codeword and rest are the received words.

The decoding table is the standard array, and the decoding rule is standard-array decoding or coset decoding. Its rows are the cosets of C, with C itself as a first row. Each is of the form e+C, where e is the coset leader. As a result, the collection of coset leaders is $\left\{0000,0101,10101,1111\right\}$.

Multiply each of them with parity-check matrix H, to get the corresponding syndromes.

coset leader	0000 1000 0100 1100
syndrome	00 11 01 10

Table 4