# Extended 1-perfect unitrades of length 10

Citation Author(s):
Denis
Krotov
Sobolev Institute of Mathematics
Submitted by:
Denis Krotov
Last updated:
Mon, 12/02/2019 - 04:04
DOI:
10.21227/86r8-ry69
Data Format:
Dataset Views:
44
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### KEYWORDS

The weight of a binary word is the number of ones in it. The N-cube is the graph whose vertices are the binary words of length N and the edges are the pairs of words differing in exactly one position. The halved N-cube is the graph whose vertices are the even-weight binary words of length N and the edjes are the pairs of words differing in exactly two positions. An extended 1-perfect unitrade is a set C of binary even-weight words of length N such that every odd-weight word is adjacent to exactly 0 or 2 words from C. Equivalently, an extended 1-perfect unitrade is a set of vertices of the halved N-cube that induces a N/2-regular subgraph without triangles. An extended 1-perfect unitrade is primary if it is non-empty and cannot be split into two non-empty extended 1-perfect unitrades.The dataset is the result of the exhaustive search of primary extended 1-perfect unitrades in the 10-cube.We list representatives of the all 38 equivalence classes (two sets A and B of binary N-words are equivlent if there is an automorphism of the N-cube that sends A to B).

The unitrades from the first 8 classes are bibartite and were described in [D.S.Krotov, The extended 1-perfect trades in small hypercubes. Discrete Math. 340(10) 2017, 2559-2572. Section 5] as extended 1-perfect bitrades.

The representatives of the equivalence classes No 0, 1, 2, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 20 are constant-weight; the other 23 classes have no constant-weight representatives. The unitrades from classes No 0, 1, 2, 4, 5, 8, 9 are reducible to unitrades of length 8.

Instructions:

In the file "uni10bin.txt", each unitrade C is represented by a line where the words from C are separated by comma.

The file "uni10dec.txt" contains the decimal representation of the same words.

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[1] Denis Krotov, "Extended 1-perfect unitrades of length 10", IEEE Dataport, 2019. [Online]. Available: http://dx.doi.org/10.21227/86r8-ry69. Accessed: Mar. 28, 2020.
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Denis Krotov. (2019). Extended 1-perfect unitrades of length 10. IEEE Dataport. http://dx.doi.org/10.21227/86r8-ry69
Denis Krotov, 2019. Extended 1-perfect unitrades of length 10. Available at: http://dx.doi.org/10.21227/86r8-ry69.
Denis Krotov. (2019). "Extended 1-perfect unitrades of length 10." Web.
1. Denis Krotov. Extended 1-perfect unitrades of length 10 [Internet]. IEEE Dataport; 2019. Available from : http://dx.doi.org/10.21227/86r8-ry69
Denis Krotov. "Extended 1-perfect unitrades of length 10." doi: 10.21227/86r8-ry69