Best Known (91−14, 91, s)-Nets in Base 7
(91−14, 91, 117657)-Net over F7 — Constructive and digital
Digital (77, 91, 117657)-net over F7, using
- (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 (70, 84, 117649)-net over F7, using
- net defined by OOA [i] based on linear OOA(784, 117649, F7, 14, 14) (dual of [(117649, 14), 1647002, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(784, 823543, F7, 14) (dual of [823543, 823459, 15]-code), using
- 1 times truncation [i] based on linear OA(785, 823544, F7, 15) (dual of [823544, 823459, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 823544 | 714−1, 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(785, 823544, F7, 15) (dual of [823544, 823459, 16]-code), using
- OA 7-folding and stacking [i] based on linear OA(784, 823543, F7, 14) (dual of [823543, 823459, 15]-code), using
- net defined by OOA [i] based on linear OOA(784, 117649, F7, 14, 14) (dual of [(117649, 14), 1647002, 15]-NRT-code), using
- digital (0, 7, 8)-net over F7, using
(91−14, 91, 823584)-Net over F7 — Digital
Digital (77, 91, 823584)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(791, 823584, F7, 14) (dual of [823584, 823493, 15]-code), using
- 1 times truncation [i] based on linear OA(792, 823585, F7, 15) (dual of [823585, 823493, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(8) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(750, 823543, F7, 9) (dual of [823543, 823493, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(77, 42, F7, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(14) ⊂ Ce(8) [i] based on
- 1 times truncation [i] based on linear OA(792, 823585, F7, 15) (dual of [823585, 823493, 16]-code), using
(91−14, 91, large)-Net in Base 7 — Upper bound on s
There is no (77, 91, large)-net in base 7, because
- 12 times m-reduction [i] would yield (77, 79, large)-net in base 7, but