Reed-Solomon Code

Description

Reed-Solomon Code is an error-correcting code used for robust data transmission in digital communications. It adds redundancy to the original data by appending extra symbols (parity symbols) to create codewords. These codewords can detect and correct errors caused by noise or data corruption during transmission. Reed-Solomon Codes are widely used in applications where reliability and data integrity are critical, such as CDs, DVDs, QR codes, and satellite communication.

C++ Code

#include <iostream>
#include <vector>
#include <algorithm>
using namespace std;

int gf256_pow(int x, int power) {
    if (power == 0) return 1;
    int result = x;
    for (int i = 2; i <= power; i++) {
        result = gf256_mul(result, x);
    }
    return result;
}

int gf256_mul(int a, int b) {
    if (a == 0 || b == 0) return 0;
    return gf256_exp[(gf256_log[a] + gf256_log[b]) % 255];
}

void gf256_init() {
    int x = 1;
    for (int i = 0; i < 255; i++) {
        gf256_exp[i] = x;
        gf256_log[x] = i;
        x = (x * 2) ^ (x >= 128 ? 0x11D : 0);
    }
}

vector<int> reedSolomonEncode(vector<int>& data, int n, int k) {
    vector<int> result(n);
    for (int i = 0; i < k; i++) {
        result[i] = data[i];
    }
    for (int i = k; i < n; i++) {
        result[i] = 0;
        for (int j = 0; j < k; j++) {
            result[i] ^= gf256_mul(data[j], gf256_pow(i, j));
        }
    }
    return result;
}

int main() {
    gf256_init();

    vector<int> data = {0, 1, 2, 3};
    int n = 6;
    int k = 4;

    vector<int> encoded = reedSolomonEncode(data, n, k);

    cout << "Encoded Reed-Solomon codeword: ";
    for (int symbol : encoded) {
        cout << symbol << " ";
    }
    cout << endl;

    return 0;
}

Time and Space Complexity

Operation	Time Complexity	Space Complexity
Reed-Solomon Encoding	O(n * k)	O(n)

Where:

• n: Length of the codeword.
• k: Number of data symbols.

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