Best Known (36, 36+31, s)-Nets in Base 81
(36, 36+31, 730)-Net over F81 — Constructive and digital
Digital (36, 67, 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, 36+31, 3290)-Net over F81 — Digital
Digital (36, 67, 3290)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8167, 3290, F81, 2, 31) (dual of [(3290, 2), 6513, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8167, 6580, F81, 31) (dual of [6580, 6513, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8167, 6581, F81, 31) (dual of [6581, 6514, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(23) [i] based on
- linear OA(8161, 6561, F81, 31) (dual of [6561, 6500, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(816, 20, F81, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,81)), using
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- Reed–Solomon code RS(75,81) [i]
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- construction X applied to Ce(30) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(8167, 6581, F81, 31) (dual of [6581, 6514, 32]-code), using
- OOA 2-folding [i] based on linear OA(8167, 6580, F81, 31) (dual of [6580, 6513, 32]-code), using
(36, 36+31, large)-Net in Base 81 — Upper bound on s
There is no (36, 67, large)-net in base 81, because
- 29 times m-reduction [i] would yield (36, 38, large)-net in base 81, but