Best Known (34−19, 34, s)-Nets in Base 81
(34−19, 34, 266)-Net over F81 — Constructive and digital
Digital (15, 34, 266)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- digital (4, 23, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- digital (2, 11, 116)-net over F81, using
(34−19, 34, 390)-Net over F81 — Digital
Digital (15, 34, 390)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8134, 390, F81, 19) (dual of [390, 356, 20]-code), using
- 89 step Varšamov–Edel lengthening with (ri) = (2, 19 times 0, 1, 68 times 0) [i] based on linear OA(8131, 298, F81, 19) (dual of [298, 267, 20]-code), using
- extended algebraic-geometric code AGe(F,278P) [i] based on function field F/F81 with g(F) = 12 and N(F) ≥ 298, using
- 89 step Varšamov–Edel lengthening with (ri) = (2, 19 times 0, 1, 68 times 0) [i] based on linear OA(8131, 298, F81, 19) (dual of [298, 267, 20]-code), using
(34−19, 34, 515746)-Net in Base 81 — Upper bound on s
There is no (15, 34, 515747)-net in base 81, because
- 1 times m-reduction [i] would yield (15, 33, 515747)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 955 011081 490701 508278 453684 791392 990481 898317 433430 440953 021041 > 8133 [i]