Best Known (81−30, 81, s)-Nets in Base 9
(81−30, 81, 344)-Net over F9 — Constructive and digital
Digital (51, 81, 344)-net over F9, using
- 7 times m-reduction [i] based on digital (51, 88, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 44, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 44, 172)-net over F81, using
(81−30, 81, 737)-Net over F9 — Digital
Digital (51, 81, 737)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(981, 737, F9, 30) (dual of [737, 656, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(981, 739, F9, 30) (dual of [739, 658, 31]-code), using
- construction XX applied to C1 = C([63,91]), C2 = C([66,92]), C3 = C1 + C2 = C([66,91]), and C∩ = C1 ∩ C2 = C([63,92]) [i] based on
- linear OA(976, 728, F9, 29) (dual of [728, 652, 30]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {63,64,…,91}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(973, 728, F9, 27) (dual of [728, 655, 28]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {66,67,…,92}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(979, 728, F9, 30) (dual of [728, 649, 31]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {63,64,…,92}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(970, 728, F9, 26) (dual of [728, 658, 27]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {66,67,…,91}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(92, 8, F9, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,9)), using
- discarding factors / shortening the dual code based on linear OA(92, 9, F9, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,9)), using
- Reed–Solomon code RS(7,9) [i]
- discarding factors / shortening the dual code based on linear OA(92, 9, F9, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,9)), using
- 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
- construction XX applied to C1 = C([63,91]), C2 = C([66,92]), C3 = C1 + C2 = C([66,91]), and C∩ = C1 ∩ C2 = C([63,92]) [i] based on
- discarding factors / shortening the dual code based on linear OA(981, 739, F9, 30) (dual of [739, 658, 31]-code), using
(81−30, 81, 114170)-Net in Base 9 — Upper bound on s
There is no (51, 81, 114171)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 196649 854778 681391 409846 717737 602881 634658 541788 933360 996486 709592 109726 423081 > 981 [i]