Best Known (73−36, 73, s)-Nets in Base 81
(73−36, 73, 730)-Net over F81 — Constructive and digital
Digital (37, 73, 730)-net over F81, using
- t-expansion [i] based on digital (36, 73, 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
(73−36, 73, 2189)-Net over F81 — Digital
Digital (37, 73, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8173, 2189, F81, 3, 36) (dual of [(2189, 3), 6494, 37]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8173, 6567, F81, 36) (dual of [6567, 6494, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(8173, 6569, F81, 36) (dual of [6569, 6496, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(32) [i] based on
- linear OA(8171, 6561, F81, 36) (dual of [6561, 6490, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(8165, 6561, F81, 33) (dual of [6561, 6496, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(35) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(8173, 6569, F81, 36) (dual of [6569, 6496, 37]-code), using
- OOA 3-folding [i] based on linear OA(8173, 6567, F81, 36) (dual of [6567, 6494, 37]-code), using
(73−36, 73, 5188076)-Net in Base 81 — Upper bound on s
There is no (37, 73, 5188077)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 20 864468 631425 899688 859145 080224 161643 477012 591425 316677 178968 113635 884957 882338 337560 763246 498082 030359 358589 001317 930839 395104 765375 336481 > 8173 [i]