Detailed explanation of BCH code and RS code

This column contains the core knowledge of information theory and coding, organized by knowledge points, and can be used as a reference for teaching or learning. The markdown version has been archived to [Github repository: https://github.com/timerring/information-theory ] or public account [AIShareLab] to reply to Information Theory .

BCH code-cyclic code

Features: There is a close relationship between its generator polynomial g(x) and the minimum code distance, and the BCH code can be easily constructed according to the required error correction capability t.

If the generator polynomial of the cyclic code has the following form:
$$g(x)=\mathrm{LCM}\left[m_1(x),m_2\left(x\right),\dots ,m_{2t-}\left(x\right)\right]$$
where t is the number of error corrections, $m_{i}(x) $ is a reduced (prime) polynomial,$LCM$ means to take the least common multiple, then the cyclic code generated by it is$BCH$ code. Code distance$\mathbf{d}\ge 2t\_+1$ . Each codeword can correct$t$ random independent errors.

if $The\; code\; length\; of\; BCH$ code$n=2^m-1$ , where$m\ge 3$, $t<\frac{m}{2}$, $n-k\le mt$ , then it is the original$BCH$ code;

if $The\; code\; length\; of\; the\; BCH$ code is$n=2^m-1$ factor, it is non-primitive$BCH$ code.

related information:

Definition of a primitive polynomial : a polynomial f(x) of degree n

(1) f(x) is a reduced polynomial (unfactorable) -GF(2);

(2) f(x) 是 $\left(x^{p+1}\right)$factor, $ p^{2}=2{n}-1$

(3) f(x) is not a factor of $ \left(x^{q+1}\right)$, $ p>q$

$Encoding\; of\; BCH$ code: generate polynomial look-up table.

$Decoding\; of\; BCH$ code:

$The\; decoding\; of\; BCH$ mainly adopts Peterson decoding, and the idea is as follows:

• Use the factors of the generator polynomial g(x) as the division formula, calculate the remainder of the received code polynomial, and obtain t remainders, which are called partial adjoint formulas;
• Construct a specific decoding polynomial with t partial syndromes, whose root is the number of error positions;
• Find the root of the decoding polynomial to get the error position;
• Correct the wrong position.

RS code

q base $A\; special\; subclass\; of\; BCH$ code (n=q-1), and has strong error correction ability.

$Parameters\; of\; RS$ code: code length n=q-1, number of supervision bits r=2t, where t is the number of error codes that can be corrected, minimum code distance d = 2 t + 1 \$\mathbf{d}=2t+1$ ; its generating polynomial is
$$g(x)=(x+a)\left(x+a^2\right)\dots \left(x+a^{2t}\right)$$
, where$a^i$ is Galois field$\mathbf{GF}\left(a^m\right)$in an element.

The main advantages of RS codes:

• It is a multi-ary system error correction code, so it is especially suitable for the occasion of multi-ary system modulation;
• It can correct t q-bit binary error codes, that is, it can correct no more than q consecutive binary error codes, so it is suitable for correcting burst error codes in fading channels.

Summarize

• Basic concepts of cyclic codes;
• Generator polynomials for cyclic codes, and encoding codes with this polynomial
• Constructing Systematic Generator Matrix and Supervisory Matrix from Generator Polynomials
• Encoding and decoding circuit of cyclic code;
• BCH, RS code.

references:

1. Proakis, John G., et al. Communication systems engineering. Vol. 2. New Jersey: Prentice Hall, 1994.
2. Proakis, John G., et al. SOLUTIONS MANUAL Communication Systems Engineering. Vol. 2. New Jersey: Prentice Hall, 1994.
3. Zhou Jiongpan. Communication Principles (3rd Edition) [M]. Beijing: Beijing University of Posts and Telecommunications Press, 2008.
4. Fan Changxin, Cao Lina. Principles of Communication (7th Edition) [M]. Beijing: National Defense Industry Press, 2012.