Best Known (68, 68+34, s)-Nets in Base 7
(68, 68+34, 200)-Net over F7 — Constructive and digital
Digital (68, 102, 200)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (17, 34, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 17, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- trace code for nets [i] based on digital (0, 17, 50)-net over F49, using
- digital (34, 68, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 34, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49 (see above)
- trace code for nets [i] based on digital (0, 34, 50)-net over F49, using
- digital (17, 34, 100)-net over F7, using
(68, 68+34, 915)-Net over F7 — Digital
Digital (68, 102, 915)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7102, 915, F7, 34) (dual of [915, 813, 35]-code), using
- 812 step Varšamov–Edel lengthening with (ri) = (6, 3, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 32 times 0, 1, 34 times 0, 1, 36 times 0, 1, 39 times 0, 1, 41 times 0, 1, 44 times 0, 1, 46 times 0, 1, 50 times 0) [i] based on linear OA(734, 35, F7, 34) (dual of [35, 1, 35]-code or 35-arc in PG(33,7)), using
- dual of repetition code with length 35 [i]
- 812 step Varšamov–Edel lengthening with (ri) = (6, 3, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 32 times 0, 1, 34 times 0, 1, 36 times 0, 1, 39 times 0, 1, 41 times 0, 1, 44 times 0, 1, 46 times 0, 1, 50 times 0) [i] based on linear OA(734, 35, F7, 34) (dual of [35, 1, 35]-code or 35-arc in PG(33,7)), using
(68, 68+34, 140717)-Net in Base 7 — Upper bound on s
There is no (68, 102, 140718)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 158 494870 447978 602048 525700 670133 274046 348631 356072 531764 715474 900388 398070 654227 909749 > 7102 [i]