Best Known (72−36, 72, s)-Nets in Base 81
(72−36, 72, 730)-Net over F81 — Constructive and digital
Digital (36, 72, 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
(72−36, 72, 2069)-Net over F81 — Digital
Digital (36, 72, 2069)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8172, 2069, F81, 3, 36) (dual of [(2069, 3), 6135, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8172, 2188, F81, 3, 36) (dual of [(2188, 3), 6492, 37]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8172, 6564, F81, 36) (dual of [6564, 6492, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(8172, 6566, F81, 36) (dual of [6566, 6494, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(33) [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(8167, 6561, F81, 34) (dual of [6561, 6494, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(8172, 6566, F81, 36) (dual of [6566, 6494, 37]-code), using
- OOA 3-folding [i] based on linear OA(8172, 6564, F81, 36) (dual of [6564, 6492, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(8172, 2188, F81, 3, 36) (dual of [(2188, 3), 6492, 37]-NRT-code), using
(72−36, 72, 4064238)-Net in Base 81 — Upper bound on s
There is no (36, 72, 4064239)-net in base 81, because
- 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]