Best Known (107−19, 107, s)-Nets in Base 9
(107−19, 107, 59065)-Net over F9 — Constructive and digital
Digital (88, 107, 59065)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 16)-net over F9, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 1 and N(F) ≥ 16, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- digital (78, 97, 59049)-net over F9, using
- net defined by OOA [i] based on linear OOA(997, 59049, F9, 19, 19) (dual of [(59049, 19), 1121834, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(997, 531442, F9, 19) (dual of [531442, 531345, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(997, 531442, F9, 19) (dual of [531442, 531345, 20]-code), using
- net defined by OOA [i] based on linear OOA(997, 59049, F9, 19, 19) (dual of [(59049, 19), 1121834, 20]-NRT-code), using
- digital (1, 10, 16)-net over F9, using
(107−19, 107, 531487)-Net over F9 — Digital
Digital (88, 107, 531487)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9107, 531487, F9, 19) (dual of [531487, 531380, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- linear OA(997, 531441, F9, 19) (dual of [531441, 531344, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(961, 531441, F9, 12) (dual of [531441, 531380, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(910, 46, F9, 6) (dual of [46, 36, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(910, 52, F9, 6) (dual of [52, 42, 7]-code), using
- a “Gra†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(910, 52, F9, 6) (dual of [52, 42, 7]-code), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
(107−19, 107, large)-Net in Base 9 — Upper bound on s
There is no (88, 107, large)-net in base 9, because
- 17 times m-reduction [i] would yield (88, 90, large)-net in base 9, but