Best Known (108−15, 108, s)-Nets in Base 7
(108−15, 108, 823564)-Net over F7 — Constructive and digital
Digital (93, 108, 823564)-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 (82, 97, 823543)-net over F7, using
- net defined by OOA [i] based on linear OOA(797, 823543, F7, 15, 15) (dual of [(823543, 15), 12353048, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(797, 5764802, F7, 15) (dual of [5764802, 5764705, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 5764802 | 716−1, 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(797, 5764802, F7, 15) (dual of [5764802, 5764705, 16]-code), using
- net defined by OOA [i] based on linear OOA(797, 823543, F7, 15, 15) (dual of [(823543, 15), 12353048, 16]-NRT-code), using
- digital (4, 11, 21)-net over F7, using
(108−15, 108, 5764855)-Net over F7 — Digital
Digital (93, 108, 5764855)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7108, 5764855, F7, 15) (dual of [5764855, 5764747, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(7105, 5764849, F7, 15) (dual of [5764849, 5764744, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(8) [i] based on
- linear OA(797, 5764801, F7, 15) (dual of [5764801, 5764704, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(757, 5764801, F7, 9) (dual of [5764801, 5764744, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(78, 48, F7, 5) (dual of [48, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(78, 50, F7, 5) (dual of [50, 42, 6]-code), using
- a “Gra†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(78, 50, F7, 5) (dual of [50, 42, 6]-code), using
- construction X applied to Ce(14) ⊂ Ce(8) [i] based on
- linear OA(7105, 5764852, F7, 14) (dual of [5764852, 5764747, 15]-code), using Gilbert–Varšamov bound and bm = 7105 > Vbs−1(k−1) = 16290 160193 567063 579736 169665 995478 282667 000900 149177 082832 870450 371358 439389 617583 616727 [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(7105, 5764849, F7, 15) (dual of [5764849, 5764744, 16]-code), using
- construction X with Varšamov bound [i] based on
(108−15, 108, large)-Net in Base 7 — Upper bound on s
There is no (93, 108, large)-net in base 7, because
- 13 times m-reduction [i] would yield (93, 95, large)-net in base 7, but