Best Known (26, 41, s)-Nets in Base 7
(26, 41, 121)-Net over F7 — Constructive and digital
Digital (26, 41, 121)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 21)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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, 8, 13)-net over F7, using
- 5 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (0, 3, 8)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (15, 30, 100)-net over F7, using
- trace code 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
- trace code for nets [i] based on digital (0, 15, 50)-net over F49, using
- digital (4, 11, 21)-net over F7, using
(26, 41, 360)-Net over F7 — Digital
Digital (26, 41, 360)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(741, 360, F7, 15) (dual of [360, 319, 16]-code), using
- 10 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0) [i] based on linear OA(738, 347, F7, 15) (dual of [347, 309, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(737, 343, F7, 15) (dual of [343, 306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(734, 343, F7, 13) (dual of [343, 309, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(71, 4, F7, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- 10 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0) [i] based on linear OA(738, 347, F7, 15) (dual of [347, 309, 16]-code), using
(26, 41, 38005)-Net in Base 7 — Upper bound on s
There is no (26, 41, 38006)-net in base 7, because
- 1 times m-reduction [i] would yield (26, 40, 38006)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 6366 879357 396681 738194 641634 674657 > 740 [i]