Best Known (109−34, 109, s)-Nets in Base 7
(109−34, 109, 344)-Net over F7 — Constructive and digital
Digital (75, 109, 344)-net over F7, using
- base reduction for projective spaces (embedding PG(54,49) in PG(108,7)) for nets [i] based on digital (21, 55, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
(109−34, 109, 1374)-Net over F7 — Digital
Digital (75, 109, 1374)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7109, 1374, F7, 34) (dual of [1374, 1265, 35]-code), using
- 1264 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, 1, 52 times 0, 1, 56 times 0, 1, 59 times 0, 1, 63 times 0, 1, 67 times 0, 1, 72 times 0, 1, 76 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]
- 1264 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, 1, 52 times 0, 1, 56 times 0, 1, 59 times 0, 1, 63 times 0, 1, 67 times 0, 1, 72 times 0, 1, 76 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
(109−34, 109, 313580)-Net in Base 7 — Upper bound on s
There is no (75, 109, 313581)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 130 527911 265758 904544 370520 409456 727329 253760 093983 961873 717734 115936 005321 192500 214394 626575 > 7109 [i]