Best Known (73−37, 73, s)-Nets in Base 81
(73−37, 73, 730)-Net over F81 — Constructive and digital
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
(73−37, 73, 1823)-Net over F81 — Digital
Digital (36, 73, 1823)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8173, 1823, F81, 3, 37) (dual of [(1823, 3), 5396, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8173, 2187, F81, 3, 37) (dual of [(2187, 3), 6488, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8173, 6561, F81, 37) (dual of [6561, 6488, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- OOA 3-folding [i] based on linear OA(8173, 6561, F81, 37) (dual of [6561, 6488, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(8173, 2187, F81, 3, 37) (dual of [(2187, 3), 6488, 38]-NRT-code), using
(73−37, 73, 4064238)-Net in Base 81 — Upper bound on s
There is no (36, 73, 4064239)-net in base 81, because
- 1 times m-reduction [i] would yield (36, 72, 4064239)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 257585 747316 136842 222287 132445 812443 836125 563373 902910 005708 380867 131249 840147 547089 269665 631592 623168 304205 520871 069002 263791 278613 080161 > 8172 [i]