Best Known (39, 39+31, s)-Nets in Base 81
(39, 39+31, 730)-Net over F81 — Constructive and digital
Digital (39, 70, 730)-net over F81, using
- t-expansion [i] based on digital (36, 70, 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+31, 5055)-Net over F81 — Digital
Digital (39, 70, 5055)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8170, 5055, F81, 31) (dual of [5055, 4985, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8170, 6591, F81, 31) (dual of [6591, 6521, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,10]) [i] based on
- linear OA(8161, 6562, F81, 31) (dual of [6562, 6501, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(819, 29, F81, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,81)), using
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- Reed–Solomon code RS(72,81) [i]
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- construction X applied to C([0,15]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8170, 6591, F81, 31) (dual of [6591, 6521, 32]-code), using
(39, 39+31, large)-Net in Base 81 — Upper bound on s
There is no (39, 70, large)-net in base 81, because
- 29 times m-reduction [i] would yield (39, 41, large)-net in base 81, but