A128036 - OEIS

%I #21 Jan 12 2020 10:05:25

%S 3,3,4,9,9,11,13,17,19

%N Maximal possible number of vectors in {0,1,2}^n such that the Hamming distance between every two is odd.

%C The sequence f_3(n) (the analog for even Hamming distances) is probably 2^(n-1) for odd values of n and that value plus 1 for even n.

%H N. Alon and E. Lubetzky, <a href="https://m.tau.ac.il/~nogaa/PDFS/xor4.pdf">Codes and XOR graph products</a>, Combinatorica, 27 (No. 1, 2007), 13-33. [See g_3(n).]

%H Fausto A. C. Cariboni, <a href="/A128036/a128036.txt">A maximal set for a(9) = 19</a>, Nov 17 2019.

%H Rahul Sarkar, Ewout van den Berg, <a href="https://arxiv.org/abs/1909.08123">On sets of commuting and anticommuting Paulis</a>, arXiv:1909.08123 [quant-ph], 2019.

%e Lexicographically earliest maximal sets:

%e a(1) = 3 {0, 1, 2}

%e a(2) = 3 {00, 01, 02}

%e a(3) = 4 {000, 001, 112, 122}

%e a(4) = 9 {0000, 0111, 0222, 1012, 1120, 1201, 2021, 2102, 2210}

%e a(5) = 9 {00000, 00111, 00222, 01012, 01120, 01201, 02021, 02102, 02210}

%e a(6) = 11 {000000, 000111, 000222, 001012, 010120, 111201, 212012, 221120, 222021, 222102, 222210}

%e a(7) = 13 {0000000, 0000111, 0000222, 0001012, 0001120, 0001201, 0002021, 0112102, 0222210, 1012210, 1202102, 2022102, 2102210}

%e a(8) = 17 {00000000, 00000111, 00000222, 00001012, 00010120, 01102012, 02220120, 10120120, 11211201, 12002012, 20202012, 21020120, 22112012, 22121120, 22122021, 22122102, 22122210}

%Y Cf. A128037.

%K nonn,nice,more

%O 1,1

%A Alon Noga (nogaa(AT)post.tau.ac.il) and Eyal Lubetzky (lubetzky(AT)post.tau.ac.il), May 03 2007

%E a(7)-a(8) from _Bert Dobbelaere_, Dec 26 2018

%E a(9) from _Fausto A. C. Cariboni_, Nov 17 2019