Best Known (80, 80+15, s)-Nets in Base 7
(80, 80+15, 117665)-Net over F7 — Constructive and digital
Digital (80, 95, 117665)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 16)-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 (0, 7, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7 (see above)
- digital (0, 3, 8)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (70, 85, 117649)-net over F7, using
- net defined by OOA [i] based on linear OOA(785, 117649, F7, 15, 15) (dual of [(117649, 15), 1764650, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [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]
- OOA 7-folding and stacking with additional row [i] based on linear OA(785, 823544, F7, 15) (dual of [823544, 823459, 16]-code), using
- net defined by OOA [i] based on linear OOA(785, 117649, F7, 15, 15) (dual of [(117649, 15), 1764650, 16]-NRT-code), using
- digital (3, 10, 16)-net over F7, using
(80, 80+15, 823591)-Net over F7 — Digital
Digital (80, 95, 823591)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(795, 823591, F7, 15) (dual of [823591, 823496, 16]-code), using
- construction X with Varšamov bound [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
- linear OA(792, 823588, F7, 14) (dual of [823588, 823496, 15]-code), using Gilbert–Varšamov bound and bm = 792 > Vbs−1(k−1) = 168216 745625 373109 205703 774249 876477 893154 370957 720507 283141 352369 217490 712151 [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
- linear OA(792, 823585, F7, 15) (dual of [823585, 823493, 16]-code), using
- construction X with Varšamov bound [i] based on
(80, 80+15, large)-Net in Base 7 — Upper bound on s
There is no (80, 95, large)-net in base 7, because
- 13 times m-reduction [i] would yield (80, 82, large)-net in base 7, but