Best Known (22, 22+32, s)-Nets in Base 81
(22, 22+32, 370)-Net over F81 — Constructive and digital
Digital (22, 54, 370)-net over F81, using
- t-expansion [i] based on digital (16, 54, 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
(22, 22+32, 373)-Net over F81 — Digital
Digital (22, 54, 373)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8154, 373, F81, 3, 32) (dual of [(373, 3), 1065, 33]-NRT-code), using
- construction X applied to AG(3;F,1074P) ⊂ AG(3;F,1081P) [i] based on
- linear OOA(8148, 369, F81, 3, 32) (dual of [(369, 3), 1059, 33]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1074P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- linear OOA(8141, 369, F81, 3, 25) (dual of [(369, 3), 1066, 26]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1081P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370 (see above)
- linear OOA(816, 4, F81, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(816, 81, F81, 3, 6) (dual of [(81, 3), 237, 7]-NRT-code), using
- Reed–Solomon NRT-code RS(3;237,81) [i]
- discarding factors / shortening the dual code based on linear OOA(816, 81, F81, 3, 6) (dual of [(81, 3), 237, 7]-NRT-code), using
- construction X applied to AG(3;F,1074P) ⊂ AG(3;F,1081P) [i] based on
(22, 22+32, 234731)-Net in Base 81 — Upper bound on s
There is no (22, 54, 234732)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 11 434177 242531 627550 863191 619238 768606 452201 479290 513617 052148 099478 436558 536353 617804 017051 842531 240961 > 8154 [i]