Blokkikodeissa

Blokkikodeissa (block codes) are a class of error-correcting codes used in digital communications and data storage to protect information by adding redundancy. In a typical block code, data symbols are grouped into blocks of k symbols and encoded into blocks of n symbols, where n is greater than or equal to k. The code rate is R = k/n, and the code is fixed-length, in contrast to convolutional codes.

Linear block codes are the most common type, described by a generator matrix G (k by n) and a parity-check matrix H ((n−k) by n). Each valid codeword c satisfies c = uG for a data vector u, and HC^T = 0. A key parameter is the minimum distance d_min, the smallest distance between distinct codewords. The error-detecting capability is up to d_min−1 errors, and the error-correcting capability is up to floor((d_min−1)/2) errors per block.

Common families include Hamming codes (short and simple), BCH codes (longer blocks with larger d_min), and Reed–Solomon codes (symbol-based codes widely used in storage and transmission). RS codes are particularly prominent in data protection for CDs, DVDs, Blu-ray discs, QR codes, and various broadcast and networking standards.

Applications cover memory error correction (Hamming and related codes), storage devices (BCH and RS in flash memories and optical discs), and data transmission systems. QR codes, for example, use Reed–Solomon codes to guard against physical damage to the symbol area.

See also: Linear block code, Generator matrix, Parity-check matrix, Minimum distance, Error correction code.