zbMATH — the first resource for mathematics

A new class of linear error-correcting codes. (Eine neue Klasse linearer fehlerkorrigierender Codes.) (Russian. English summary) Zbl 0292.94011
Probl. Peredaci Inform. 6, No. 3, 24-30 (1970); translation in Probl. Inf. Transm. 6, No. 3, 207-212 (1970).
Let \(L= \{\alpha_1,\ldots,\alpha_n\} \subseteq \mathrm{GF}(2^m)\) \(n\le 2^m)\) be a nonempty set. For every \(x = (x_1,\ldots,x_n)\) we define a rational function \(R_x(z) = \sum_{i=1}^n (x_i/(z - \alpha_i))\). Let \(g(z)\) be a polynomial with coefficients from \(\mathrm{GF}(2^m)\) having no roots in \(L\). The binary \((L,g)\)-code (Goppa-code) \(C\) is the set of all \(x\) with \(R_x(z)\equiv 0 \bmod g(z)\). Obviously this code is linear. There are at most \(m\cdot\deg g(z)\) check symbols in \(C\). Let \(\bar g(z)\) be a polynomial of minimum degree which is a complete square with \(g(z)\mid \bar g(z)\). Then for the minimum distance \(d\) in \(C\) holds \(d\ge \deg \bar g(z) + 1\). Decoding is similar to Peterson’s algorithm for BCH-codes. If \(C\) is cyclic then it is an BCH-code. For example a \((16,8,5)\)-\((L,g)\)-code is constructed.

94B05 Linear codes (general theory)
94B15 Cyclic codes
94B35 Decoding
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
Full Text: MNR