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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 $$C$$.

##### MSC:
 94B05 Linear codes (general theory) 05A19 Combinatorial identities, bijective combinatorics 20N05 Loops, quasigroups
##### Keywords:
combinatorial polarization; binary linear code
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