Best Known (64−28, 64, s)-Nets in Base 81
(64−28, 64, 730)-Net over F81 — Constructive and digital
Digital (36, 64, 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
(64−28, 64, 5542)-Net over F81 — Digital
Digital (36, 64, 5542)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8164, 5542, F81, 28) (dual of [5542, 5478, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8164, 6590, F81, 28) (dual of [6590, 6526, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(17) [i] based on
- 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(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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 Ce(27) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(8164, 6590, F81, 28) (dual of [6590, 6526, 29]-code), using
(64−28, 64, large)-Net in Base 81 — Upper bound on s
There is no (36, 64, large)-net in base 81, because
- 26 times m-reduction [i] would yield (36, 38, large)-net in base 81, but