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Combinatorial aspects of code loops. (English) Zbl 1042.94023
For a mapping \(P: V \to F\), \(P(0) = 0\), where \(V\) is a vector space over \(F = \{0,1\}\), define \(P_r: V^r \to F\), \(r \geq 1\), as \(P_r(v_1,\dots ,v_r) = \sum P(\sum _{i \in B}v_i)\), where \(B\) runs through all non-empty subsets of \(\{1,\dots ,r\}\). The combinatorial degree of \(P\) is the least \(r\geq 0\) with \(P_{r+1} = 0\). The level of a binary code \(C\) is the least \(r\) with \(2^r\) dividing \(| u| \) for all \(u \in C\). The main result of the paper is as follows: Let \(P\) be of combinatorial degree \(r+1\). Then \(V\) can be identified with a binary code \(C\) of level \(r\) in such a way that \(P(c) = | c| /2^r\) for every \(c \in C\). The result is obtained by describing a construction of \(C\).

94B05 Linear codes (general theory)
05A19 Combinatorial identities, bijective combinatorics
20N05 Loops, quasigroups
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