Best Known (54, 85, s)-Nets in Base 9
(54, 85, 344)-Net over F9 — Constructive and digital
Digital (54, 85, 344)-net over F9, using
- 9 times m-reduction [i] based on digital (54, 94, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 47, 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, 47, 172)-net over F81, using
(54, 85, 781)-Net over F9 — Digital
Digital (54, 85, 781)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(985, 781, F9, 31) (dual of [781, 696, 32]-code), using
- 41 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0, 1, 32 times 0) [i] based on linear OA(983, 738, F9, 31) (dual of [738, 655, 32]-code), using
- construction XX applied to C1 = C([726,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([726,28]) [i] based on
- 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 = {−2,−1,…,27}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(976, 728, F9, 29) (dual of [728, 652, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(982, 728, F9, 31) (dual of [728, 646, 32]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {−2,−1,…,28}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(973, 728, F9, 28) (dual of [728, 655, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(91, 7, F9, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- Reed–Solomon code RS(8,9) [i]
- discarding factors / shortening the dual code based on linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), 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([726,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([726,28]) [i] based on
- 41 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0, 1, 32 times 0) [i] based on linear OA(983, 738, F9, 31) (dual of [738, 655, 32]-code), using
(54, 85, 177179)-Net in Base 9 — Upper bound on s
There is no (54, 85, 177180)-net in base 9, because
- 1 times m-reduction [i] would yield (54, 84, 177180)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 143 349753 324400 244580 355964 021186 379794 971625 298318 917298 316281 178442 921677 055649 > 984 [i]