Best Known (21−7, 21, s)-Nets in Base 9
(21−7, 21, 400)-Net over F9 — Constructive and digital
Digital (14, 21, 400)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 212)-net over F9, using
- net defined by OOA [i] based on linear OOA(95, 212, F9, 3, 3) (dual of [(212, 3), 631, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(95, 212, F9, 2, 3) (dual of [(212, 2), 419, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(95, 212, F9, 3, 3) (dual of [(212, 3), 631, 4]-NRT-code), using
- digital (9, 16, 200)-net over F9, using
- trace code for nets [i] based on digital (1, 8, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 8, 100)-net over F81, using
- digital (2, 5, 212)-net over F9, using
(21−7, 21, 878)-Net over F9 — Digital
Digital (14, 21, 878)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(921, 878, F9, 7) (dual of [878, 857, 8]-code), using
- 142 step Varšamov–Edel lengthening with (ri) = (1, 20 times 0, 1, 120 times 0) [i] based on linear OA(919, 734, F9, 7) (dual of [734, 715, 8]-code), using
- construction XX applied to C1 = C([727,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([727,5]) [i] based on
- linear OA(916, 728, F9, 6) (dual of [728, 712, 7]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,4}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(916, 728, F9, 6) (dual of [728, 712, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(919, 728, F9, 7) (dual of [728, 709, 8]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(913, 728, F9, 5) (dual of [728, 715, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,4], 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([727,4]), C2 = C([0,5]), C3 = C1 + C2 = C([0,4]), and C∩ = C1 ∩ C2 = C([727,5]) [i] based on
- 142 step Varšamov–Edel lengthening with (ri) = (1, 20 times 0, 1, 120 times 0) [i] based on linear OA(919, 734, F9, 7) (dual of [734, 715, 8]-code), using
(21−7, 21, 522287)-Net in Base 9 — Upper bound on s
There is no (14, 21, 522288)-net in base 9, because
- 1 times m-reduction [i] would yield (14, 20, 522288)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 12 157724 102748 591745 > 920 [i]