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Modular curves and codes with a polynomial construction. (English) Zbl 0544.94016

Let \(F_ q\) be a field of q elements. Then a q-ary block code of length n is an arbitrary subset \(C\subseteq F^ n_ q\). Hamming distance between a and b, \(a\in F^ n_ q\), \(b\in F^{n}_{q}\) is the number of unequal components in a and b. The rate and the relative distance of the code are \(R(C)=\log_ q| C| /n\); \(\delta(C)=d(C)/n\), where d(C) is the minimal Hamming distance between two unequal vectors in C. Let \(I=\{1,...,| C| \}\). A bijective map \(\phi:I\to C\) is referred to as coding. The code C has a polynomial complexity of construction iff we can construct \(\phi\) with polynomial complexity in n and \(\phi\) (i) itself also has polynomial complexity in n. The problem is to find such codes having maximal possible rate (\(\delta\) (C) being fixed). In this paper we prove a new lower bound for the rate of binary codes with polynomial complexity of construction, which is better than the best known bound [see Eh. L. Blokh and V. V. Zyablov, Linear concatenated codes (Russian) (1982)]. We use the construction of concatenated codes with the fixed binary codes as the inner codes and codes arising from the classical modular curves through the Goppa construction as outer codes.

MSC:

94B05 Linear codes (general theory)
14H45 Special algebraic curves and curves of low genus
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