Best Known (42, 77, s)-Nets in Base 81
(42, 77, 730)-Net over F81 — Constructive and digital
Digital (42, 77, 730)-net over F81, using
- t-expansion [i] based on digital (36, 77, 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
(42, 77, 4073)-Net over F81 — Digital
Digital (42, 77, 4073)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8177, 4073, F81, 35) (dual of [4073, 3996, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(8177, 6587, F81, 35) (dual of [6587, 6510, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(25) [i] based on
- linear OA(8169, 6561, F81, 35) (dual of [6561, 6492, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(818, 26, F81, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,81)), using
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- Reed–Solomon code RS(73,81) [i]
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- construction X applied to Ce(34) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(8177, 6587, F81, 35) (dual of [6587, 6510, 36]-code), using
(42, 77, large)-Net in Base 81 — Upper bound on s
There is no (42, 77, large)-net in base 81, because
- 33 times m-reduction [i] would yield (42, 44, large)-net in base 81, but