Best Known (37, 37+40, s)-Nets in Base 81
(37, 37+40, 730)-Net over F81 — Constructive and digital
Digital (37, 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
(37, 37+40, 1215)-Net over F81 — Digital
Digital (37, 77, 1215)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8177, 1215, F81, 40) (dual of [1215, 1138, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(8177, 1312, F81, 40) (dual of [1312, 1235, 41]-code), using
(37, 37+40, 2311435)-Net in Base 81 — Upper bound on s
There is no (37, 77, 2311436)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 898 150718 363950 245969 827247 589981 445802 999035 351343 557623 991143 426229 554077 052754 005003 602859 973942 488688 104135 032273 020165 003496 247831 316087 033601 > 8177 [i]