Best Known (59, 59+30, s)-Nets in Base 7
(59, 59+30, 150)-Net over F7 — Constructive and digital
Digital (59, 89, 150)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (14, 29, 50)-net over F7, using
- base reduction for projective spaces (embedding PG(14,49) in PG(28,7)) for nets [i] based on digital (0, 15, 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
- base reduction for projective spaces (embedding PG(14,49) in PG(28,7)) for nets [i] based on digital (0, 15, 50)-net over F49, using
- digital (30, 60, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 30, 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, 30, 50)-net over F49, using
- digital (14, 29, 50)-net over F7, using
(59, 59+30, 778)-Net over F7 — Digital
Digital (59, 89, 778)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(789, 778, F7, 30) (dual of [778, 689, 31]-code), using
- 688 step Varšamov–Edel lengthening with (ri) = (6, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 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, 7 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, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 36 times 0, 1, 38 times 0, 1, 42 times 0, 1, 44 times 0, 1, 48 times 0) [i] based on linear OA(730, 31, F7, 30) (dual of [31, 1, 31]-code or 31-arc in PG(29,7)), using
- dual of repetition code with length 31 [i]
- 688 step Varšamov–Edel lengthening with (ri) = (6, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 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, 7 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, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 36 times 0, 1, 38 times 0, 1, 42 times 0, 1, 44 times 0, 1, 48 times 0) [i] based on linear OA(730, 31, F7, 30) (dual of [31, 1, 31]-code or 31-arc in PG(29,7)), using
(59, 59+30, 110618)-Net in Base 7 — Upper bound on s
There is no (59, 89, 110619)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1635 966803 937409 320357 228060 648791 838888 684902 680788 726106 080372 105035 217223 > 789 [i]