BCH Codes

Abstract

In this chapter, we will give information about Bose-Chaudhuri-Hocquenghem, i.e., BCH, and Reed-Solomon codes. These codes fall into the category of linear cyclic codes. BCH codes are binary cyclic codes, and on the other hand, Reed-Solomon codes are nonbinary cyclic codes. The generator polynomials of these cyclic codes are constructed using the minimal polynomials of the extended finite fields. For this reason, it is vital to have fundamental knowledge of extended finite fields to comprehend the topics presented. In this chapter, we first explain the construction of the generator polynomials of the BCH codes using minimal polynomials of the extended fields. Using the generator polynomials, generator and parity check matrices of the BCH codes can be obtained. Next, the syndrome decoding operation of the BCH codes using error location polynomial is explained by examples. Finally, we provide information about the PGZ algorithm used for the decoding of BCH codes.

Problems

• 1. Obtain the generator polynomial of the single-error-correcting BCH code. Use GF(2³) which is constructed using the primitive polynomial p(x) = x³ + x² + 1.

• 2. Obtain the generator polynomial of the double-error-correcting BCH code. Use GF(2⁴) which is constructed using the primitive polynomial p(x) = x⁴ + x³ + 1.

• 3. Obtain the generator polynomial of the triple-error-correcting BCH code. Use GF(2⁵) which is constructed using the primitive polynomial p(x) = x⁵ + x³ + 1.

• 4. Find the generator and parity check matrices of the double-error-correcting BCH code obtained using GF(2⁴). Use p(x) = x⁴ + x³ + 1 for the construction of GF(2⁴).

• 5. Find the generator and parity check matrices of the double-error-correcting BCH code. Use GF(2⁵) constructed using the primitive polynomial p(x) = x⁵ + x³ + 1.

• 6. The extended field GF(2⁴) is constructed using the primitive polynomial p(x) = x⁴ + x + 1. The generator polynomial of the triple-error-correcting BCH(15, 5) code over GF(2⁴) is evaluated as

  $$ g(x)={x}^{10}+{x}^8+{x}^5+{x}^4+{x}^2+x+1. $$

• Find the parity check and generator matrices of the dual code.

• 7. The generator polynomial of the double-error-correcting BCH(15, 7) cyclic code is given as

  $$ g(x)={x}^8+{x}^7+{x}^6+{x}^4+1. $$

• The data-word polynomial is d(x) = x⁶ + x³ + x + 1. Obtain the systematic and non-systematic code-words for d(x).

• 8. For any triple-error-correcting BCH code, find the coefficients of error location polynomial in terms of the syndromes.

• 9. The triple-error-correcting BCH(15, 5) code is constructed on the extended field GF(2⁴). The extended field GF(2⁴) is obtained using the primitive polynomial p(x) = x⁴ + x³ + 1.

  The data-word polynomial is d(x) = x⁶ + x³ + x + 1. Assume that a data-word is encoded and the generated code-word c(x) is transmitted. The received word at the receiver side is expressed in polynomial form as

  $$ r(x)=c(x)+e(x) $$

• where e(x) = x⁴ + x². Using r(x), determine the transmitted code-word using the PGZ algorithm.