Coding theory

Coding theory studies mathematical methods for representing information and detecting or correcting errors in storage and transmission. It balances reliability, data rate, and the resources needed to encode and decode messages.

Hamming distance: The number of positions at which two strings of equal length differ. Minimum distance determines how many errors a code can detect or correct.
Finite field: A field containing finitely many elements, with addition, subtraction, multiplication, and division by nonzero elements. Many algebraic codes perform their calculations over finite fields.
Compact disc: An optical disc format for storing and retrieving digital audio and other data. Interleaved Reed–Solomon codes help recover data despite scratches and dust.
Data compression: The representation of data using fewer bits than its original encoding. Compression removes redundancy, while error-correcting codes deliberately add it.
Parity-check matrix: A matrix whose null space defines a linear code and whose product with a received word yields its syndrome. Its syndrome identifies whether a received codeword violates the code's constraints.
Linear code: A set of vectors over a field that forms a vector subspace. Linearity gives coding theory compact descriptions and efficient encoding and decoding methods.
QR code: A two-dimensional barcode that stores data in a grid of dark and light modules. Reed–Solomon error correction allows decoding when parts of the symbol are obscured.
Cryptography: The study and practice of securing information through mathematical techniques. Cryptography protects confidentiality or authenticity; coding theory protects against accidental corruption.
Syndrome decoding: A decoding method that uses parity-check results to infer likely transmission errors. It maps detected constraint violations to error patterns for correction.
Generator matrix: A matrix whose rows form a basis for a linear code. Multiplying information vectors by it produces valid codewords.

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