Best Known (38, 38+39, s)-Nets in Base 81
(38, 38+39, 730)-Net over F81 — Constructive and digital
Digital (38, 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
(38, 38+39, 1868)-Net over F81 — Digital
Digital (38, 77, 1868)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8177, 1868, F81, 3, 39) (dual of [(1868, 3), 5527, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8177, 2187, F81, 3, 39) (dual of [(2187, 3), 6484, 40]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8177, 6561, F81, 39) (dual of [6561, 6484, 40]-code), using
- an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- OOA 3-folding [i] based on linear OA(8177, 6561, F81, 39) (dual of [6561, 6484, 40]-code), using
- discarding factors / shortening the dual code based on linear OOA(8177, 2187, F81, 3, 39) (dual of [(2187, 3), 6484, 40]-NRT-code), using
(38, 38+39, 4266430)-Net in Base 81 — Upper bound on s
There is no (38, 77, 4266431)-net in base 81, because
- 1 times m-reduction [i] would yield (38, 76, 4266431)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 11 088244 582578 440216 118969 874628 326810 474918 268083 070098 862766 514931 622250 602145 451641 535026 877542 501901 759268 196042 888990 824580 853280 329070 165521 > 8176 [i]