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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.

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