BCH algorithm

BCH (Bose-Chaudhuri-Hocquenghem) is a cyclic error-correcting code used in digital communications and storage.

Characteristics:
1. Block Code Properties:

• Can detect and correct multiple bit errors
• Good for random error correction
• Efficient hardware implementation
• Defined over Galois Field mathematics

2. Key Parameters:

• (n,k,t) where:
  • n = block length (total bits)
  • k = data length (information bits)
  • t = number of correctable errors
• Common configurations: BCH(15,7), BCH(31,16), BCH(63,45)

Main Applications:
1. Digital Storage:

• Hard disk drives
• Solid state drives
• Memory systems

2. Communications:

• Satellite communications
• Mobile networks
• Deep space communications

Comparison with Reed-Solomon:

• BCH works at bit level (RS works at symbol level)
• Better for random errors (RS better for burst errors)
• Generally simpler to implement
• Lower overhead for small block sizes

golang 3rd party package

Go doesn't have built-in support for BCH in its standard library, but there are some third-party implementations available.

1. github.com/klauspost/reedsolomon - While primarily for Reed-Solomon, it includes some BCH functionality
2. Some hardware-focused Go libraries that include BCH implementations for FPGA programming

For production use, you might want to consider:
1. Using an existing tested library
2. Implementing the full BCH algorithm including:

• Proper generator polynomial calculation
• Syndrome calculation
• Error location polynomial calculation
• Chien search for finding error locations
• Complete error correction

package main

import (
    "fmt"
)

// GaloisField represents operations in GF(2^m)
type GaloisField struct {
    m      uint    // Field degree
    n      uint    // Field size (2^m - 1)
    alpha  []uint  // Powers of primitive element
    log    []uint  // Logarithm table
}

// BCHEncoder represents a BCH encoder
type BCHEncoder struct {
    n   uint      // Code length
    k   uint      // Message length
    t   uint      // Error correction capability
    g   []uint    // Generator polynomial
    gf  *GaloisField
}

func NewBCHEncoder(m, t uint) *BCHEncoder {
    // Initialize Galois Field
    gf := initGaloisField(m)
    
    // Calculate generator polynomial
    // This is simplified - real implementation would need complete polynomial calculation
    n := uint(1<<m) - 1
    k := n - m*t
    
    return &BCHEncoder{
        n:  n,
        k:  k,
        t:  t,
        gf: gf,
    }
}

func (bch *BCHEncoder) Encode(data []byte) []byte {
    // Encoding implementation would go here
    // Returns encoded data with parity bits
    return nil
}

func (bch *BCHEncoder) Decode(received []byte) ([]byte, error) {
    // Decoding implementation would go here
    // Returns corrected data and any error
    return nil, nil
}

// Example usage
func main() {
    // Create a BCH(63,51) code that can correct up to 2 errors
    bch := NewBCHEncoder(6, 2)
    
    // Example data
    data := []byte{0x55, 0xAA, 0x55, 0xAA, 0x55, 0xAA}
    
    // Encode
    encoded := bch.Encode(data)
    fmt.Printf("Encoded: %x\n", encoded)
    
    // Simulate transmission with errors
    received := encoded
    // Add some errors here...
    
    // Decode
    decoded, err := bch.Decode(received)
    if err != nil {
        fmt.Printf("Error: %v\n", err)
    }
    fmt.Printf("Decoded: %x\n", decoded)
}

BCH and Reed-solomon

In satellite communications, both BCH and Reed-Solomon have their specific uses in TC (telecommand) error correction, but they serve different purposes in the protocol stack.

For TC Transfer Frame error correction:
1. BCH is typically used for:

• Error correction within code blocks of CLTU (Command Link Transmission Unit)
• Specifically, BCH(63,56) is commonly used according to CCSDS standards
• Provides bit-level error correction for individual code blocks

2. Reed-Solomon is typically used for:

• Higher layer error correction
• Longer block protection
• Usually when dealing with larger data units

The choice depends on:
1. Layer requirements:

• Physical layer often uses BCH for immediate error correction
• Transfer layer might use RS for additional protection

2. Error patterns:

• BCH better for random bit errors in shorter blocks
• RS better for burst errors and longer blocks

3. CCSDS Standards:

• Different recommendations for different layers
• Some standards specifically mandate BCH for CLTU code blocks
• RS might be recommended for other layers