Best Known (18−7, 18, s)-Nets in Base 9
(18−7, 18, 246)-Net over F9 — Constructive and digital
Digital (11, 18, 246)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 82)-net over F9, using
- net defined by OOA [i] based on linear OOA(94, 82, F9, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
- digital (7, 14, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 7, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 7, 82)-net over F81, using
- digital (1, 4, 82)-net over F9, using
(18−7, 18, 370)-Net over F9 — Digital
Digital (11, 18, 370)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(918, 370, F9, 7) (dual of [370, 352, 8]-code), using
- construction XX applied to C1 = C([41,46]), C2 = C([40,45]), C3 = C1 + C2 = C([41,45]), and C∩ = C1 ∩ C2 = C([40,46]) [i] based on
- linear OA(915, 364, F9, 6) (dual of [364, 349, 7]-code), using the BCH-code C(I) with length 364 | 93−1, defining interval I = {41,42,…,46}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(915, 364, F9, 6) (dual of [364, 349, 7]-code), using the BCH-code C(I) with length 364 | 93−1, defining interval I = {40,41,…,45}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(918, 364, F9, 7) (dual of [364, 346, 8]-code), using the BCH-code C(I) with length 364 | 93−1, defining interval I = {40,41,…,46}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(912, 364, F9, 5) (dual of [364, 352, 6]-code), using the BCH-code C(I) with length 364 | 93−1, defining interval I = {41,42,43,44,45}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(90, 3, F9, 0) (dual of [3, 3, 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(90, 3, F9, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([41,46]), C2 = C([40,45]), C3 = C1 + C2 = C([41,45]), and C∩ = C1 ∩ C2 = C([40,46]) [i] based on
(18−7, 18, 58030)-Net in Base 9 — Upper bound on s
There is no (11, 18, 58031)-net in base 9, because
- 1 times m-reduction [i] would yield (11, 17, 58031)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 16677 453719 801449 > 917 [i]