Best Known (89, 89+15, s)-Nets in Base 7
(89, 89+15, 823551)-Net over F7 — Constructive and digital
Digital (89, 104, 823551)-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 (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 (0, 7, 8)-net over F7, using
(89, 89+15, 4688468)-Net over F7 — Digital
Digital (89, 104, 4688468)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7104, 4688468, F7, 15) (dual of [4688468, 4688364, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(7104, 5764810, F7, 15) (dual of [5764810, 5764706, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([1,7]) [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]
- linear OA(796, 5764802, F7, 7) (dual of [5764802, 5764706, 8]-code), using the narrow-sense BCH-code C(I) with length 5764802 | 716−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(77, 8, F7, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,7)), using
- dual of repetition code with length 8 [i]
- construction X applied to C([0,7]) ⊂ C([1,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(7104, 5764810, F7, 15) (dual of [5764810, 5764706, 16]-code), using
(89, 89+15, large)-Net in Base 7 — Upper bound on s
There is no (89, 104, large)-net in base 7, because
- 13 times m-reduction [i] would yield (89, 91, large)-net in base 7, but