Best Known (39, 39+37, s)-Nets in Base 81
(39, 39+37, 730)-Net over F81 — Constructive and digital
Digital (39, 76, 730)-net over F81, using
- t-expansion [i] based on digital (36, 76, 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
(39, 39+37, 2394)-Net over F81 — Digital
Digital (39, 76, 2394)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8176, 2394, F81, 2, 37) (dual of [(2394, 2), 4712, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8176, 3286, F81, 2, 37) (dual of [(3286, 2), 6496, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8176, 6572, F81, 37) (dual of [6572, 6496, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(8176, 6573, F81, 37) (dual of [6573, 6497, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(8173, 6562, F81, 37) (dual of [6562, 6489, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(8165, 6562, F81, 33) (dual of [6562, 6497, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [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 C([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8176, 6573, F81, 37) (dual of [6573, 6497, 38]-code), using
- OOA 2-folding [i] based on linear OA(8176, 6572, F81, 37) (dual of [6572, 6496, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(8176, 3286, F81, 2, 37) (dual of [(3286, 2), 6496, 38]-NRT-code), using
(39, 39+37, large)-Net in Base 81 — Upper bound on s
There is no (39, 76, large)-net in base 81, because
- 35 times m-reduction [i] would yield (39, 41, large)-net in base 81, but