7.2 Error-Locator Polynomials

Error-locator polynomials are key to decoding BCH codes. They help pinpoint where errors occur in received codewords. By finding the roots of these polynomials, we can figure out exactly which bits got messed up during transmission.

This topic builds on earlier concepts in BCH codes. It shows how we use mathematical tools like polynomials and Galois fields to detect and fix errors. Understanding this process is crucial for grasping how BCH codes work in practice.

Error-Locator Polynomial and Syndrome Polynomial

Defining the Error-Locator Polynomial

• Error-locator polynomial $\Lambda(z)$ is a polynomial whose roots are the reciprocals of the error locations in a received codeword
• Defined as $\Lambda(z) = \prod_{i=1}^{v}(1-X_iz)$ where $X_i$ are the error locations and $v$ is the number of errors
• Coefficients of $\Lambda(z)$ provide information about the error locations in the received codeword
• Finding the roots of $\Lambda(z)$ allows for determining the positions of the errors in the codeword

Syndrome Polynomial and Equations

• Syndrome polynomial $S(z)$ is a polynomial derived from the syndromes of a received codeword
• Syndromes are calculated by evaluating the received polynomial at the roots of the generator polynomial
• Syndrome equations are a set of equations relating the syndromes and the coefficients of the error-locator polynomial
• Solving the syndrome equations helps determine the coefficients of $\Lambda(z)$
• Example syndrome equation: $S_1\Lambda_1 + S_2\Lambda_0 = S_3$

Error Pattern and Correction

• Error pattern refers to the positions and values of the errors in a received codeword
• Once the error-locator polynomial is determined, the error pattern can be found
• Error correction involves subtracting the error pattern from the received codeword to obtain the original transmitted codeword
• Example: If the error pattern is $(0, e_1, 0, e_3)$, the errors are at positions 1 and 3 with values $e_1$ and $e_3$

Error Locations and Values

Determining Error Locations

• Error location numbers are the positions of the errors in the received codeword
• Found by calculating the roots of the error-locator polynomial $\Lambda(z)$
• Roots are typically expressed as powers of the primitive element $\alpha$ of the Galois field
• Example: If the roots of $\Lambda(z)$ are $\alpha^2$ and $\alpha^5$, the errors are at positions 2 and 5

Calculating Error Values

• Error values are the actual values of the errors at the error locations
• Calculated using Forney's algorithm, which involves evaluating a modified syndrome polynomial at the error locations
• Modified syndrome polynomial is obtained by dividing the syndrome polynomial by the error-locator polynomial
• Example: If the error location is $\alpha^3$ and the modified syndrome polynomial evaluated at $\alpha^3$ is $\beta$, the error value is $\beta$

Newton's Identities and Decoding

• Newton's identities are a set of equations relating the coefficients of a polynomial to its power sums
• Used in the decoding process to solve for the coefficients of the error-locator polynomial
• Involve the syndromes and the elementary symmetric functions of the error locations
• Solving Newton's identities helps determine the error-locator polynomial, leading to the error locations and values
• Example: $S_1 = \Lambda_1$, $S_2 = \Lambda_1S_1 - 2\Lambda_2$, $S_3 = \Lambda_1S_2 - \Lambda_2S_1 + 3\Lambda_3$