Best Known (19, 19+21, s)-Nets in Base 81
(19, 19+21, 370)-Net over F81 — Constructive and digital
Digital (19, 40, 370)-net over F81, using
- t-expansion [i] based on digital (16, 40, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(19, 19+21, 811)-Net over F81 — Digital
Digital (19, 40, 811)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8140, 811, F81, 21) (dual of [811, 771, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8140, 1312, F81, 21) (dual of [1312, 1272, 22]-code), using
(19, 19+21, 1570278)-Net in Base 81 — Upper bound on s
There is no (19, 40, 1570279)-net in base 81, because
- 1 times m-reduction [i] would yield (19, 39, 1570279)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 269 722921 880793 924284 848434 114928 464596 641216 015115 717736 735235 976223 903201 > 8139 [i]