Kódhosszt

Kódhosszt is a term used in coding theory to refer to the length of a code, i.e., the number of symbols in each codeword. In mathematical notation, the code length is usually denoted by n. For a code C over an alphabet of size q, codewords are elements of the vector space F_q^n, and every codeword has length n. If the code is linear, it is commonly described by parameters [n, k, d], where n is the code length, k the dimension, and d the minimum distance.

The code length is a fixed characteristic of a given code and together with the dimension k determines the code rate R = k/n, which measures information density. The minimum distance d, in combination with n, governs the error-detecting and error-correcting capabilities of the code: in general, a code can detect up to d−1 errors and correct up to floor((d−1)/2) errors per codeword under appropriate decoding.

Longer codes (larger n) can offer greater absolute error-correcting power for a fixed q, but may require more redundancy or higher decoding complexity. Several bounds relate the parameters n, k, d, and q, including the Singleton bound, Hamming bound, and Gilbert–Varshamov bound, highlighting trade-offs among length, rate, and reliability.

Common examples illustrate the concept: a binary Hamming code has length n = 2^r − 1, and Reed–Solomon codes have length n ≤ q − 1 over the finite field GF(q). In practice, the code length affects latency in data transmission and storage systems, making it a central design parameter in coding schemes.

Etymology: in Hungarian, kódhosszt is the accusative form of kódhossz, meaning code length. See also code length, linear code, minimum distance, and error correction.