Question

Construct a standard array for the (6,3) code in Example in 7 of Section 16.1 and find the syndrome of each coset leader.

Short answer

The set of coset leaders is

$\left\{000000,001110,010101,011011,100011,101101,110110,111000\right\}$, and the corresponding set of syndromes are as $\left\{000,011,101,110,100,010,001,111\right\}$.

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 (6,3) code:

$\left\{000000,001110,010101,011011,100011,101101,11011,111000\right\}$.

The generator matrix is;

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

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;

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

The corresponding parity-check matrix is;

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

From the elements of B(6) not in C, choose one of the smallest weight (which in this case is weight 1), say $e_1=100000$. From its coset e₁+C by adding e₁ successively to the elements of C and list the coset elements, with e₁+C directly below for each data-custom-editor="chemistry" $\mathrm{c}\in \mathrm{C}$(as in table 1).

C	000000	001110 010101 011011 100011 101101
$e_1+C$	100000	101110 110101 111011 000011 001101

Table 1

Completing the table with all elements of B(6) 

Repeat until every elements of B(6) is on the table. This table is complete (as in table 2).

C	000000	001110 010101 011011 100011 101101 111000
e1+C e2+C e3+C e4+C e5+C e6+C e7+C	100000 010000 001000 000100 000010 000001 100100	101110 110101 111011 000011 010110 011000 011110 000101 001011 110011 100110 101000 000110 011101 010011 101011 111110 110000 001010 010001 011111 100111 110010 111100 001100 010111 011001 100001 110100 111010 001111 010100 011010 100010 110111 111001 101010 110001 111111 000111 010010 011100

Table 2

Finding corresponding syndromes

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 is the coset leader. As a result, the collection of coset leaders is $\left\{000000,001110,010101,011011,100011,101101,110110,111000\right\}$.

Multiply each of them with parity-check matrix H, to get the corresponding syndromes (as in table 3).

Leader of cosets	000000 100000 010000 001000 000100 000010 000001 100100
Corresponding syndromes	000 011 101 110 100 010 001 111

Table3

Therefore, set of coset leaders is obtained as:$\left\{000000,001110,010101,011011,100011,101101,110110,111000\right\}$, and the corresponding set of syndromes are obtained as: $\left\{000,011,101,110,100,010,001,111\right\}$.