Reed-Solomon Codes

Abstract

In this chapter, we will give information about Reed-Solomon codes. These codes fall into the category of nonbinary cyclic codes. The generator polynomials of Reed-Solomon codes are constructed using the minimal polynomials of the extended finite fields. Reed-Solomon codes are effective for burst errors and they are used for erasure decoding. Reed-Solomon codes are used in some electronic devices such as CDs, DVDs, and Blu-ray, and they are also employed in communication technologies such as DSL, WiMAX, or RAID 6. Reed-Solomon codes are invented in 1960, and they are seen as nonbinary BCH codes. In this chapter, we first explain the construction of the generator polynomials of the Reed-Solomon codes using the minimal polynomials of the extended finite fields. Next, we provide information about the syndrome decoding of Reed-Solomon codes using error evaluator polynomial. In sequel, Berlekamp algorithm which is a low-complexity syndrome decoding algorithm used for the decoding of Reed-Solomon codes is explained.

Problems

• 1. The extended field GF(2³) is generated using the primitive polynomial p(x) = x³ + x² + 1. Find the generator polynomial of the single-error-correcting Reed-Solomon code over GF(2³).

• 2. Express the data vector d = [α³ 0 α⁴ α⁵ α²] in bit vector form. The elements of the data vector are chosen from GF(2³) which is constructed using the primitive polynomial p(x) = x³ + x² + 1.

• 3. Encode the data vector d = [α³ α² α α⁴ α²] using the generator polynomial of the Reed-Solomon code RS(7, 5) designed in problem 1.

• 4. Construct the generator polynomial of three-error-correcting Reed-Solomon code over GF(2⁴) which is obtained using the primitive polynomial p(x) = x⁴ + x² + 1.

• 5. The generator polynomial of the double-error-correcting Reed-Solomon code RS(7, 3) over GF(2³), constructed using the primitive polynomial p(x) = x³ + x + 1, can be calculated as

$$ g(x)={x}^4+{\alpha}^3{x}^3+{x}^2+\alpha x+{\alpha}^3. $$

The data-word polynomial is given as d(x) = α³x² + α⁵x + α⁴. Systematically encode d(x).

• 6. Using the elements of GF(2⁴), the generator polynomial of the triple-error-correcting Reed-Solomon code, RS(15, 9), is obtained. For the construction of GF(2⁴), the primitive polynomial p(x) = x⁴ + x² + 1 is used. Assume that the data-word to be encoded is given as

$$ d(x)={x}^8+{\alpha}^6{x}^5+{\alpha}^3{x}^2+x+1. $$

Encode d(x), and let c(x) be the code-word obtained after encoding operation. The error pattern is given as e(x) = α²x⁹ + α⁵x⁴. Decode the received word polynomial r(x) = c(x) + e(x). Use Berlekamp algorithm in your decoding operation.