7.1 BCH Code Construction and Parameters

BCH codes are powerful error-correcting codes built using finite fields. They use a generator polynomial with specific roots to create codewords. The code's strength comes from its designed distance, which determines how many errors it can fix.

Cyclotomic cosets play a key role in BCH code construction. They help find the roots for the generator polynomial. The code's parameters, like error-correction ability and rate, depend on these roots and the code's design.

BCH Code Fundamentals

Key Components of BCH Codes

Top images from around the web for Key Components of BCH Codes

• 

  Half Rate Quasi Cyclic Low Density Parity Check Codes Based on Combinatorial Designs View original

  Is this image relevant?

• 

  galois theory - Finding all elements in GF(2^4) in terms of given polynomial - Mathematics Stack ... View original

  Is this image relevant?

• 

  arithmetic operations in Galois Field - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Half Rate Quasi Cyclic Low Density Parity Check Codes Based on Combinatorial Designs View original

  Is this image relevant?

• 

  galois theory - Finding all elements in GF(2^4) in terms of given polynomial - Mathematics Stack ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Key Components of BCH Codes

• 

  Half Rate Quasi Cyclic Low Density Parity Check Codes Based on Combinatorial Designs View original

  Is this image relevant?

• 

  galois theory - Finding all elements in GF(2^4) in terms of given polynomial - Mathematics Stack ... View original

  Is this image relevant?

• 

  arithmetic operations in Galois Field - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Half Rate Quasi Cyclic Low Density Parity Check Codes Based on Combinatorial Designs View original

  Is this image relevant?

• 

  galois theory - Finding all elements in GF(2^4) in terms of given polynomial - Mathematics Stack ... View original

  Is this image relevant?

1 of 3

• BCH codes are a class of cyclic error-correcting codes named after Bose, Chaudhuri, and Hocquenghem
• Constructed using finite fields, also known as Galois fields, which are fields with a finite number of elements (GF(2^m))
• Galois fields contain a primitive element, denoted as $\alpha$, which generates all non-zero elements of the field when raised to successive powers
• The generator polynomial $g(x)$ is used to generate the codewords of a BCH code and is defined as the lowest degree polynomial over the Galois field that has $\alpha$, $\alpha^2$, ..., $\alpha^{2t}$ as roots, where $t$ is the error-correcting capability of the code

Cyclotomic Cosets in BCH Code Construction

• Cyclotomic cosets are used to determine the roots of the generator polynomial in BCH code construction
• A cyclotomic coset $C_s$ is a set of integers obtained by repeatedly multiplying $s$ by 2 modulo $n$, where $n = 2^m - 1$ and $m$ is the degree of the Galois field
• The elements of a cyclotomic coset are the powers of $\alpha$ that are roots of the generator polynomial
• The generator polynomial $g(x)$ is the product of the minimal polynomials of the elements in the chosen cyclotomic cosets

BCH Code Parameters

Error Correction and Code Distance

• The designed distance $d$ of a BCH code is the number of consecutive powers of $\alpha$ (starting from $\alpha$) that are roots of the generator polynomial
• The minimum distance $d_{min}$ is the smallest Hamming distance between any two distinct codewords in the code
• The error-correcting capability $t$ of a BCH code is related to the designed distance by $t = \lfloor \frac{d-1}{2} \rfloor$, where $\lfloor \cdot \rfloor$ denotes the floor function
• The BCH bound states that the minimum distance of a BCH code is at least as large as its designed distance, i.e., $d_{min} \geq d$

Code Rate and Efficiency

• The code rate $R$ of a BCH code is the ratio of the number of information bits $k$ to the total number of bits in a codeword $n$, i.e., $R = \frac{k}{n}$
• A higher code rate indicates that more information bits are transmitted per codeword, resulting in better bandwidth efficiency
• However, a higher code rate also implies a lower error-correcting capability, as fewer redundant bits are available for error correction
• The choice of code rate depends on the specific application requirements, such as the desired error performance and available bandwidth

Polynomial Roots in BCH Codes

Conjugate Roots and Their Significance

• In BCH codes, if $\alpha^i$ is a root of the generator polynomial, then its conjugate roots $\alpha^{2i}$, $\alpha^{4i}$, ..., $\alpha^{2^{m-1}i}$ are also roots of the generator polynomial, where $m$ is the degree of the Galois field
• The presence of conjugate roots in the generator polynomial ensures that the resulting BCH code is cyclic
• Cyclic codes have the property that any cyclic shift of a codeword is also a codeword, which simplifies encoding and decoding operations
• The inclusion of conjugate roots in the generator polynomial also helps in achieving the desired error-correcting capability and minimum distance of the BCH code

Primitive and Minimal Polynomials

• A primitive polynomial is a monic irreducible polynomial over a finite field that has a primitive element as one of its roots
• The minimal polynomial of an element $\alpha^i$ in a Galois field is the lowest degree monic polynomial with coefficients from the base field that has $\alpha^i$ as a root
• In BCH code construction, the generator polynomial is formed by taking the product of the minimal polynomials of the chosen roots (elements of the cyclotomic cosets)
• The degree of the minimal polynomial of an element $\alpha^i$ is equal to the size of the cyclotomic coset containing $i$