Best Known (76−36, 76, s)-Nets in Base 81
(76−36, 76, 730)-Net over F81 — Constructive and digital
Digital (40, 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
(76−36, 76, 3117)-Net over F81 — Digital
Digital (40, 76, 3117)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8176, 3117, F81, 2, 36) (dual of [(3117, 2), 6158, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8176, 3289, F81, 2, 36) (dual of [(3289, 2), 6502, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8176, 6578, F81, 36) (dual of [6578, 6502, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(29) [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(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to Ce(35) ⊂ Ce(29) [i] based on
- OOA 2-folding [i] based on linear OA(8176, 6578, F81, 36) (dual of [6578, 6502, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(8176, 3289, F81, 2, 36) (dual of [(3289, 2), 6502, 37]-NRT-code), using
(76−36, 76, large)-Net in Base 81 — Upper bound on s
There is no (40, 76, large)-net in base 81, because
- 34 times m-reduction [i] would yield (40, 42, large)-net in base 81, but