Best Known (40, 63, s)-Nets in Base 7
(40, 63, 121)-Net over F7 — Constructive and digital
Digital (40, 63, 121)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (6, 17, 21)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (1, 12, 13)-net over F7, using
- 1 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (0, 5, 8)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (23, 46, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 50)-net over F49, using
- digital (6, 17, 21)-net over F7, using
(40, 63, 410)-Net over F7 — Digital
Digital (40, 63, 410)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(763, 410, F7, 23) (dual of [410, 347, 24]-code), using
- 59 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 27 times 0) [i] based on linear OA(758, 346, F7, 23) (dual of [346, 288, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(758, 343, F7, 23) (dual of [343, 285, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(755, 343, F7, 22) (dual of [343, 288, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(70, 3, F7, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 59 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 9 times 0, 1, 17 times 0, 1, 27 times 0) [i] based on linear OA(758, 346, F7, 23) (dual of [346, 288, 24]-code), using
(40, 63, 47432)-Net in Base 7 — Upper bound on s
There is no (40, 63, 47433)-net in base 7, because
- 1 times m-reduction [i] would yield (40, 62, 47433)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 24893 313200 628231 952800 920361 429971 571787 088793 615127 > 762 [i]