Best Known (36, 69, s)-Nets in Base 81
(36, 69, 730)-Net over F81 — Constructive and digital
Digital (36, 69, 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
(36, 69, 2740)-Net over F81 — Digital
Digital (36, 69, 2740)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8169, 2740, F81, 2, 33) (dual of [(2740, 2), 5411, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8169, 3287, F81, 2, 33) (dual of [(3287, 2), 6505, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8169, 6574, F81, 33) (dual of [6574, 6505, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8169, 6575, F81, 33) (dual of [6575, 6506, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(27) [i] based on
- linear OA(8165, 6561, F81, 33) (dual of [6561, 6496, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(814, 14, F81, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(32) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(8169, 6575, F81, 33) (dual of [6575, 6506, 34]-code), using
- OOA 2-folding [i] based on linear OA(8169, 6574, F81, 33) (dual of [6574, 6505, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(8169, 3287, F81, 2, 33) (dual of [(3287, 2), 6505, 34]-NRT-code), using
(36, 69, large)-Net in Base 81 — Upper bound on s
There is no (36, 69, large)-net in base 81, because
- 31 times m-reduction [i] would yield (36, 38, large)-net in base 81, but