Best Known (103−14, 103, s)-Nets in Base 7
(103−14, 103, 823551)-Net over F7 — Constructive and digital
Digital (89, 103, 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, 96, 823543)-net over F7, using
- net defined by OOA [i] based on linear OOA(796, 823543, F7, 14, 14) (dual of [(823543, 14), 11529506, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(796, 5764801, F7, 14) (dual of [5764801, 5764705, 15]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(797, 5764802, F7, 15) (dual of [5764802, 5764705, 16]-code), using
- OA 7-folding and stacking [i] based on linear OA(796, 5764801, F7, 14) (dual of [5764801, 5764705, 15]-code), using
- net defined by OOA [i] based on linear OOA(796, 823543, F7, 14, 14) (dual of [(823543, 14), 11529506, 15]-NRT-code), using
- digital (0, 7, 8)-net over F7, using
(103−14, 103, 5764845)-Net over F7 — Digital
Digital (89, 103, 5764845)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7103, 5764845, F7, 14) (dual of [5764845, 5764742, 15]-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(76, 44, F7, 4) (dual of [44, 38, 5]-code), using
- a “GraX†code from Grassl’s database [i]
- construction X applied to Ce(14) ⊂ Ce(8) [i] based on
(103−14, 103, large)-Net in Base 7 — Upper bound on s
There is no (89, 103, large)-net in base 7, because
- 12 times m-reduction [i] would yield (89, 91, large)-net in base 7, but