Best Known (77−38, 77, s)-Nets in Base 81
(77−38, 77, 730)-Net over F81 — Constructive and digital
Digital (39, 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
(77−38, 77, 2189)-Net over F81 — Digital
Digital (39, 77, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8177, 2189, F81, 3, 38) (dual of [(2189, 3), 6490, 39]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8177, 6567, F81, 38) (dual of [6567, 6490, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(8177, 6569, F81, 38) (dual of [6569, 6492, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(34) [i] based on
- linear OA(8175, 6561, F81, 38) (dual of [6561, 6486, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- 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(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(37) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(8177, 6569, F81, 38) (dual of [6569, 6492, 39]-code), using
- OOA 3-folding [i] based on linear OA(8177, 6567, F81, 38) (dual of [6567, 6490, 39]-code), using
(77−38, 77, 5376645)-Net in Base 81 — Upper bound on s
There is no (39, 77, 5376646)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 898 145569 592605 914803 285456 369128 549622 658320 501263 663840 617997 356616 219017 359535 495378 196197 054762 791076 587722 225355 798610 069659 246363 378054 556321 > 8177 [i]