Syndrome Decoding and Some Important Linear Block Codes

Abstract

In this chapter, we will first explain the decoding of linear block codes using syndrome tables, then provide information about some well-known preliminary binary linear block codes. Syndrome decoding is a preliminary decoding approach; most of the modern decoding techniques do not employ syndrome decoding. However, to comprehend the decoding logic of linear block codes, it is essential to understand the syndrome decoding operation. For this purpose, we first explain the construction of standard arrays and decoding operation using standard arrays. Syndromes are nothing but concise representation of standard arrays. After standard arrays, we explain the syndrome decoding concept. Following the explanation of the syndrome decoding, we introduce some linear block codes well-known in the literature, such as Golay code, Reed-Muller codes, and Hamming codes, and discuss the error correction capability and decoding operations of these linear block codes. Non-systematic forms of the generator and parity check matrices of the Hamming codes are also explained.

Problems

• 1. The generator matrix of a binary linear block code is given as

$$ \boldsymbol{G}=\left[\begin{array}{c}100110\\ {}110011\\ {}101101\end{array}\right]. $$

• Find all the code-words, and construct the standard table for this code.

• 2. The generator matrix of a binary linear block code is given as

$$ \boldsymbol{G}=\left[\begin{array}{c}1001100\\ {}1100111\\ {}1111011\end{array}\right]. $$

• Determine the standard array of the dual code.

• 3. The generator matrix of a binary linear block code is given as

$$ \boldsymbol{G}=\left[\begin{array}{c}11010101\\ {}01001010\\ {}10111010\end{array}\right]. $$

• Obtain the syndrome table of this code.

• 4. The parity check matrix of a single-error-correcting binary linear block code is given as

$$ \boldsymbol{H}=\left[\begin{array}{c}0010111\\ {}0101011\\ {}1001101\end{array}\right]. $$

• (a) Obtain the syndrome table of this code.

• (b) Find the generator matrix of this code, and encode the data-word d = [1011].

• (c) For part (b), assume that during the transmission, a single-bit error occurs, and the error pattern is given as e = [0010]. Determine the decoder’s estimate using the syndrome table.

• 5. Find the generator and parity check matrices of the Reed-Muller code for m = 4. Determine the minimum distance of this code.

• 6. Determine the generator and parity check matrices of the Hamming code for r = 3, and obtain the parity check matrix of the extended Hamming code.

• 7. Syndromes are noting but the linear combinations of the transposed columns of the parity check matrix according to the error patterns. Determine whether this statement is correct or not.

• 8. What is the minimum distance of a Golay code?

• 9. The generator matrix of a linear block code C(7, 3) is given as

$$ \boldsymbol{G}=\left[\begin{array}{c}11100100\\ {}01011010\\ {}11101101\end{array}\right]. $$

• (a) Determine the systematic form of the given generator matrix.

• (b) Find the parity check matrix.

• (c) Find the minimum distance of the code.

• (d) Decide on the error detection and correction capability of this code.

• (e) Construct the syndrome table of this code.

• (f) Extend this code to a C(8, 3) linear block code. Find the generator and parity check matrices of the extended code.

• 10. The parity check matrix of a linear block code C(7, 4) is given as

$$ \boldsymbol{H}=\left[\begin{array}{c}1110100\\ {}0111010\\ {}1101001\end{array}\right]. $$

• (a) Find the generator matrix in systematic form.

• (b) Find the minimum distance of the code using the parity check matrix.

• (c) Decide on the error detection and correction capability of this code.

• (d) Let t_c be the number of bit errors that the code can correct. Determine the number of words inside a Hamming sphere of radius t_c; at the center of the sphere, a code-word exists.

• (e) Construct the syndrome table of this code.

• (f) Extend this code to a C(8, 4) linear block code. Find the generator and parity check matrices of the extended code, and besides, find the minimum distance of the extended code.

• (g) Decide on the error detection and correction capability of the extended code.

• (h) Verify the singleton and Hamming bounds for this code and its extended version.