Best Known (18, 18+23, s)-Nets in Base 81
(18, 18+23, 370)-Net over F81 — Constructive and digital
Digital (18, 41, 370)-net over F81, using
- t-expansion [i] based on digital (16, 41, 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
(18, 18+23, 420)-Net over F81 — Digital
Digital (18, 41, 420)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8141, 420, F81, 23) (dual of [420, 379, 24]-code), using
- 48 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 45 times 0) [i] based on linear OA(8139, 370, F81, 23) (dual of [370, 331, 24]-code), using
- extended algebraic-geometric code AGe(F,346P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- 48 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 45 times 0) [i] based on linear OA(8139, 370, F81, 23) (dual of [370, 331, 24]-code), using
(18, 18+23, 534398)-Net in Base 81 — Upper bound on s
There is no (18, 41, 534399)-net in base 81, because
- 1 times m-reduction [i] would yield (18, 40, 534399)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 21847 516184 051300 601108 027692 354717 193797 831572 731825 911410 588475 343471 871121 > 8140 [i]