Best Known (53, 53+27, s)-Nets in Base 7
(53, 53+27, 200)-Net over F7 — Constructive and digital
Digital (53, 80, 200)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (13, 26, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 50)-net over F49, using
- digital (27, 54, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 50)-net over F49, using
- digital (13, 26, 100)-net over F7, using
(53, 53+27, 714)-Net over F7 — Digital
Digital (53, 80, 714)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(780, 714, F7, 27) (dual of [714, 634, 28]-code), using
- 633 step Varšamov–Edel lengthening with (ri) = (5, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 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, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 24 times 0, 1, 26 times 0, 1, 28 times 0, 1, 30 times 0, 1, 33 times 0, 1, 35 times 0, 1, 39 times 0, 1, 41 times 0, 1, 45 times 0, 1, 49 times 0) [i] based on linear OA(727, 28, F7, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,7)), using
- dual of repetition code with length 28 [i]
- 633 step Varšamov–Edel lengthening with (ri) = (5, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 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, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 24 times 0, 1, 26 times 0, 1, 28 times 0, 1, 30 times 0, 1, 33 times 0, 1, 35 times 0, 1, 39 times 0, 1, 41 times 0, 1, 45 times 0, 1, 49 times 0) [i] based on linear OA(727, 28, F7, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,7)), using
(53, 53+27, 129069)-Net in Base 7 — Upper bound on s
There is no (53, 80, 129070)-net in base 7, because
- 1 times m-reduction [i] would yield (53, 79, 129070)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 5 791248 222197 777866 057609 014995 194781 326310 620544 162419 088588 357629 > 779 [i]