Best Known (68−15, 68, s)-Nets in Base 7
(68−15, 68, 2409)-Net over F7 — Constructive and digital
Digital (53, 68, 2409)-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 (46, 61, 2401)-net over F7, using
- net defined by OOA [i] based on linear OOA(761, 2401, F7, 15, 15) (dual of [(2401, 15), 35954, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(761, 16808, F7, 15) (dual of [16808, 16747, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16808 | 710−1, 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(761, 16808, F7, 15) (dual of [16808, 16747, 16]-code), using
- net defined by OOA [i] based on linear OOA(761, 2401, F7, 15, 15) (dual of [(2401, 15), 35954, 16]-NRT-code), using
- digital (0, 7, 8)-net over F7, using
(68−15, 68, 16839)-Net over F7 — Digital
Digital (53, 68, 16839)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(768, 16839, F7, 15) (dual of [16839, 16771, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(8) [i] based on
- linear OA(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(736, 16807, F7, 9) (dual of [16807, 16771, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(77, 32, F7, 5) (dual of [32, 25, 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
(68−15, 68, large)-Net in Base 7 — Upper bound on s
There is no (53, 68, large)-net in base 7, because
- 13 times m-reduction [i] would yield (53, 55, large)-net in base 7, but