Cyclic Codes

Abstract

Cyclic codes are special type of linear block codes such that any cyclic shift of a code-word results in another code-word, and this property is called the cyclic property. Cyclic codes are easier to manipulate considering other linear block codes, and they are more preferred in practical communication systems considering the other linear block codes. Polynomials can be utilized for the characterization of cyclic codes, and this enables the cyclic codes to be analyzed analytically, and they can be constructed in an algebraic manner. For the design of a cyclic code, it is essential to determine the generator polynomial of the cyclic code. In this chapter, we will explain the construction of cyclic codes along with their encoding and decoding operations. For this purpose, we give information about determination of the generator polynomials of the cyclic codes and explain the systematic and non-systematic encoding of cyclic codes. In sequel, matrix representations of the generator and parity check polynomials of the cyclic codes are described.

Problems

• 1. The polynomial x⁹ + 1 can be factorized as

$$ {x}^9+1=\left(x+1\right)\left({x}^2+x+1\right)\left({x}^6+{x}^3+1\right). $$

• The generator polynomial of a cyclic code with block length n = 9 is given as

$$ g(x)={x}^6+{x}^3+1. $$

• (a) Determine the value of k.

• (b) Find the parity check polynomial and generator polynomial of the dual cyclic code.

• (c) Find the generator and parity check matrices of this cyclic code.

• (d) Find the minimum distance of this code.

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

• 2. The generator polynomial of C(n = 8, k = 4) cyclic code is given as

$$ g(x)={x}^4+1. $$

• (a) Find the parity check polynomial of this code.

• (b) Express the data vector d = [1 0 1 1] in polynomial form, and encode the data polynomial using non-systematic and systematic encoding methods. Obtain the systematic and non-systematic code-word and determine the locations of both data and parity bits in each code-word.

• 3. A cyclic code is used to encode a data polynomial, and the code-word

$$ c(x)={x}^5+{x}^4+1 $$

is obtained. What can be the parameters of the cyclic code used, and determine a generator matrix for this code. After determining the generator polynomial, find the data-word polynomial which yields the given code-word after encoding operation.

• 4. Using the factorization

$$ {x}^{15}+1=\left(x+1\right)\left({x}^2+x+1\right)\left({x}^4+x+1\right)\left({x}^4+{x}^3+{x}^2+x+1\right) $$

• (a) Determine the number of cyclic codes C(n = 15, k).

• (b) Find the generator polynomials of the cyclic codes C(n = 15, k = 11).

• (c) Find the generator polynomial, parity check polynomial, generator matrix, and parity check matrix of the cyclic code C(n = 15, k = 7).