On binary codes of order 3
We present a class of binary (not necessarily linear) codes containing perfect codes, Preparata codes, BCH codes of lengtht 22m+1-1, Hadamard code of length 11, l-error-correcting uniformly packed codes and also some other classes presenting remarkable regularity properties. The points of a code C of order 3 are scattered in the ambiant Hamming space so that for any point x the number of paths of length 3 joining x to the code only depends on the fact that the Hamming distance d(x, C) from x to C is 0, 1 or is greater than 1.
In the linear case the set of columns of a parity check matrix of a code of order 3 is a triple-sum-set , which is a natural extension of partial difference sets [7, 15, 2] and the orthogonal of such a code admits at most three non-zero weights.
The aim of this communication is to introduce and characterize codes of order 3 and of order 3-star in the binary case and to give some examples. The general q-any case is treated in
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