Best Known (27, 27+14, s)-Nets in Base 7
(27, 27+14, 150)-Net over F7 — Constructive and digital
Digital (27, 41, 150)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 50)-net over F7, using
- base reduction for projective spaces (embedding PG(6,49) in PG(12,7)) for nets [i] based on digital (0, 7, 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(6,49) in PG(12,7)) for nets [i] based on digital (0, 7, 50)-net over F49, using
- digital (14, 28, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 50)-net over F49, using
- digital (6, 13, 50)-net over F7, using
(27, 27+14, 448)-Net over F7 — Digital
Digital (27, 41, 448)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(741, 448, F7, 14) (dual of [448, 407, 15]-code), using
- 100 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0, 1, 30 times 0, 1, 49 times 0) [i] based on linear OA(736, 343, F7, 14) (dual of [343, 307, 15]-code), using
- 1 times truncation [i] based on linear OA(737, 344, F7, 15) (dual of [344, 307, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 344 | 76−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(737, 344, F7, 15) (dual of [344, 307, 16]-code), using
- 100 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0, 1, 30 times 0, 1, 49 times 0) [i] based on linear OA(736, 343, F7, 14) (dual of [343, 307, 15]-code), using
(27, 27+14, 50187)-Net in Base 7 — Upper bound on s
There is no (27, 41, 50188)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 44573 815708 188057 800567 824964 143777 > 741 [i]