Best Known (38, 38+36, s)-Nets in Base 81
(38, 38+36, 730)-Net over F81 — Constructive and digital
Digital (38, 74, 730)-net over F81, using
- t-expansion [i] based on digital (36, 74, 730)-net over F81, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- the Hermitian function field over F81 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
(38, 38+36, 2384)-Net over F81 — Digital
Digital (38, 74, 2384)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8174, 2384, F81, 2, 36) (dual of [(2384, 2), 4694, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8174, 3286, F81, 2, 36) (dual of [(3286, 2), 6498, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8174, 6572, F81, 36) (dual of [6572, 6498, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(31) [i] based on
- linear OA(8171, 6561, F81, 36) (dual of [6561, 6490, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(8163, 6561, F81, 32) (dual of [6561, 6498, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to Ce(35) ⊂ Ce(31) [i] based on
- OOA 2-folding [i] based on linear OA(8174, 6572, F81, 36) (dual of [6572, 6498, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(8174, 3286, F81, 2, 36) (dual of [(3286, 2), 6498, 37]-NRT-code), using
(38, 38+36, 6622675)-Net in Base 81 — Upper bound on s
There is no (38, 74, 6622676)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 1690 021385 578635 117908 591589 074264 923661 012184 343540 414265 728064 163197 218349 050094 087311 958226 601656 076546 488326 217997 239538 926804 661176 064641 > 8174 [i]