Best Known (63−27, 63, s)-Nets in Base 81
(63−27, 63, 730)-Net over F81 — Constructive and digital
Digital (36, 63, 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
(63−27, 63, 6593)-Net over F81 — Digital
Digital (36, 63, 6593)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8163, 6593, F81, 27) (dual of [6593, 6530, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(15) [i] based on
- linear OA(8153, 6561, F81, 27) (dual of [6561, 6508, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(8110, 32, F81, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,81)), using
- discarding factors / shortening the dual code based on linear OA(8110, 81, F81, 10) (dual of [81, 71, 11]-code or 81-arc in PG(9,81)), using
- Reed–Solomon code RS(71,81) [i]
- discarding factors / shortening the dual code based on linear OA(8110, 81, F81, 10) (dual of [81, 71, 11]-code or 81-arc in PG(9,81)), using
- construction X applied to Ce(26) ⊂ Ce(15) [i] based on
(63−27, 63, large)-Net in Base 81 — Upper bound on s
There is no (36, 63, large)-net in base 81, because
- 25 times m-reduction [i] would yield (36, 38, large)-net in base 81, but