Best Known (168−52, 168, s)-Nets in Base 8
(168−52, 168, 1026)-Net over F8 — Constructive and digital
Digital (116, 168, 1026)-net over F8, using
- t-expansion [i] based on digital (114, 168, 1026)-net over F8, using
- 4 times m-reduction [i] based on digital (114, 172, 1026)-net over F8, using
- trace code for nets [i] based on digital (28, 86, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 86, 513)-net over F64, using
- 4 times m-reduction [i] based on digital (114, 172, 1026)-net over F8, using
(168−52, 168, 2703)-Net over F8 — Digital
Digital (116, 168, 2703)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8168, 2703, F8, 52) (dual of [2703, 2535, 53]-code), using
- 2534 step Varšamov–Edel lengthening with (ri) = (8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 32 times 0, 1, 34 times 0, 1, 35 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 41 times 0, 1, 44 times 0, 1, 45 times 0, 1, 47 times 0, 1, 49 times 0, 1, 52 times 0, 1, 54 times 0, 1, 56 times 0, 1, 58 times 0, 1, 61 times 0, 1, 64 times 0, 1, 66 times 0, 1, 69 times 0, 1, 72 times 0, 1, 75 times 0, 1, 79 times 0, 1, 82 times 0, 1, 85 times 0, 1, 89 times 0, 1, 92 times 0, 1, 97 times 0, 1, 101 times 0, 1, 105 times 0) [i] based on linear OA(852, 53, F8, 52) (dual of [53, 1, 53]-code or 53-arc in PG(51,8)), using
- dual of repetition code with length 53 [i]
- 2534 step Varšamov–Edel lengthening with (ri) = (8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 32 times 0, 1, 34 times 0, 1, 35 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 41 times 0, 1, 44 times 0, 1, 45 times 0, 1, 47 times 0, 1, 49 times 0, 1, 52 times 0, 1, 54 times 0, 1, 56 times 0, 1, 58 times 0, 1, 61 times 0, 1, 64 times 0, 1, 66 times 0, 1, 69 times 0, 1, 72 times 0, 1, 75 times 0, 1, 79 times 0, 1, 82 times 0, 1, 85 times 0, 1, 89 times 0, 1, 92 times 0, 1, 97 times 0, 1, 101 times 0, 1, 105 times 0) [i] based on linear OA(852, 53, F8, 52) (dual of [53, 1, 53]-code or 53-arc in PG(51,8)), using
(168−52, 168, 1031665)-Net in Base 8 — Upper bound on s
There is no (116, 168, 1031666)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 52 375061 464346 909224 413109 505020 890693 049067 743965 562960 412901 797291 992806 993780 380111 667812 024560 196575 020921 318718 728723 781652 321155 498577 108186 800240 > 8168 [i]