Best Known (87, 106, s)-Nets in Base 9
(87, 106, 59059)-Net over F9 — Constructive and digital
Digital (87, 106, 59059)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 10)-net over F9, using
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 0 and N(F) ≥ 10, using
- the rational function field F9(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 9)-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 (0, 9, 10)-net over F9, using
(87, 106, 531482)-Net over F9 — Digital
Digital (87, 106, 531482)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9106, 531482, F9, 19) (dual of [531482, 531376, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9104, 531478, F9, 19) (dual of [531478, 531374, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [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(967, 531441, F9, 13) (dual of [531441, 531374, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(97, 37, F9, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(9104, 531480, F9, 18) (dual of [531480, 531376, 19]-code), using Gilbert–Varšamov bound and bm = 9104 > Vbs−1(k−1) = 136 337624 831161 448362 005612 235322 875092 079314 504022 616837 598118 105615 078081 767008 252949 492945 806841 [i]
- linear OA(90, 2, F9, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9104, 531478, F9, 19) (dual of [531478, 531374, 20]-code), using
- construction X with Varšamov bound [i] based on
(87, 106, large)-Net in Base 9 — Upper bound on s
There is no (87, 106, large)-net in base 9, because
- 17 times m-reduction [i] would yield (87, 89, large)-net in base 9, but