Best Known (91−26, 91, s)-Nets in Base 7
(91−26, 91, 221)-Net over F7 — Constructive and digital
Digital (65, 91, 221)-net over F7, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 13, 21)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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, 9, 13)-net over F7, using
- 4 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (0, 4, 8)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (13, 26, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 13, 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, 13, 50)-net over F49, using
- digital (26, 52, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 26, 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, 26, 50)-net over F49, using
- digital (5, 13, 21)-net over F7, using
(91−26, 91, 2397)-Net over F7 — Digital
Digital (65, 91, 2397)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(791, 2397, F7, 26) (dual of [2397, 2306, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(791, 2412, F7, 26) (dual of [2412, 2321, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- linear OA(789, 2401, F7, 26) (dual of [2401, 2312, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(781, 2401, F7, 24) (dual of [2401, 2320, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(71, 10, F7, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(25) ⊂ Ce(23) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(791, 2412, F7, 26) (dual of [2412, 2321, 27]-code), using
(91−26, 91, 777923)-Net in Base 7 — Upper bound on s
There is no (65, 91, 777924)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 80154 594534 802636 274799 907200 996638 286811 103739 354062 547908 373918 097012 902073 > 791 [i]