Best Known (48, 69, s)-Nets in Base 7
(48, 69, 208)-Net over F7 — Constructive and digital
Digital (48, 69, 208)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 7, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (10, 20, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 10, 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, 10, 50)-net over F49, using
- digital (21, 42, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 21, 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, 21, 50)-net over F49, using
- digital (0, 7, 8)-net over F7, using
(48, 69, 1150)-Net over F7 — Digital
Digital (48, 69, 1150)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(769, 1150, F7, 21) (dual of [1150, 1081, 22]-code), using
- 1080 step Varšamov–Edel lengthening with (ri) = (4, 2, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 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, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 10 times 0, 1, 10 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 23 times 0, 1, 25 times 0, 1, 28 times 0, 1, 31 times 0, 1, 34 times 0, 1, 38 times 0, 1, 42 times 0, 1, 46 times 0, 1, 52 times 0, 1, 57 times 0, 1, 63 times 0, 1, 69 times 0, 1, 77 times 0, 1, 85 times 0, 1, 94 times 0, 1, 104 times 0) [i] based on linear OA(721, 22, F7, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,7)), using
- dual of repetition code with length 22 [i]
- 1080 step Varšamov–Edel lengthening with (ri) = (4, 2, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 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, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 10 times 0, 1, 10 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 23 times 0, 1, 25 times 0, 1, 28 times 0, 1, 31 times 0, 1, 34 times 0, 1, 38 times 0, 1, 42 times 0, 1, 46 times 0, 1, 52 times 0, 1, 57 times 0, 1, 63 times 0, 1, 69 times 0, 1, 77 times 0, 1, 85 times 0, 1, 94 times 0, 1, 104 times 0) [i] based on linear OA(721, 22, F7, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,7)), using
(48, 69, 421197)-Net in Base 7 — Upper bound on s
There is no (48, 69, 421198)-net in base 7, because
- 1 times m-reduction [i] would yield (48, 68, 421198)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 2928 707892 548611 106413 205979 642076 292569 880734 816660 698301 > 768 [i]