On shifting sets in the binary hypercube

The extended 1-perfect trades in small hypercubes

“…In case (i), each element of a trade (T 0 , T 1 ) is self-complementary, but T 0 does not uniquely define T 1 and vice versa; T 0 and T 1 can be nonequivalent in this case (see Section 6.1 for examples). In fact, for every n ≡ 1 mod 4, there exists a 3-way 1-perfect trade [30]. For k > 3, k-way trades can also exist.…”

“…In the theory of 1-perfect codes, trades play an important role for the construction of codes with different properties and the evaluation of their number. There are not so many works where the class of 1-perfect trades is studied independently [23], [30], [31]; however, the subsets of 1-perfect codes called i-components, or switching components, which are essentially a special kind of trade mates, are used in many constructions of such codes, see the surveys in [2], [9], [20], [25], [26].…”

Cardinality spectra of components of correlation immune functions, bent functions, perfect colorings, and codes

“…Paper [7] uses the notion of a shifting set in E n , which is a union of two codes C 1 and C 2 with distance 3 having the same neighborhood. Define a function h : E n → Q by h = χ C 1 − χ C 2 .…”

“…Cardinalities of components of perfect codes or cardinalities of intersections of perfect codes were considered in [4][5][6][7][8][9]; the minimal cardinality of a component of a perfect code has been known for long, as well as the minimal cardinality of a component of a bent function (see also [10]). However, the problem of existence of components of intermediate cardinality between the minimum and twice the minimum cardinality remained little studied.…”

Embedding in a perfect code

“…With this approach, a large number of 1-perfect codes can be produced [9]. 1-Codes that admit replacing by other 1-codes with the same neighborhood deserve the independent study; they exist in F n for every odd n [20].…”