Best Known (109−85, 109, s)-Nets in Base 4
(109−85, 109, 34)-Net over F4 — Constructive and digital
Digital (24, 109, 34)-net over F4, using
- t-expansion [i] based on digital (21, 109, 34)-net over F4, using
- net from sequence [i] based on digital (21, 33)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 21 and N(F) ≥ 34, using
- T5 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 21 and N(F) ≥ 34, using
- net from sequence [i] based on digital (21, 33)-sequence over F4, using
(109−85, 109, 35)-Net in Base 4 — Constructive
(24, 109, 35)-net in base 4, using
- net from sequence [i] based on (24, 34)-sequence in base 4, using
- base expansion [i] based on digital (48, 34)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 41, N(F) = 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F2 with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
- base expansion [i] based on digital (48, 34)-sequence over F2, using
(109−85, 109, 49)-Net over F4 — Digital
Digital (24, 109, 49)-net over F4, using
- net from sequence [i] based on digital (24, 48)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 24 and N(F) ≥ 49, using
(109−85, 109, 103)-Net in Base 4 — Upper bound on s
There is no (24, 109, 104)-net in base 4, because
- 15 times m-reduction [i] would yield (24, 94, 104)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(494, 104, S4, 70), but
- the linear programming bound shows that M ≥ 2905 343984 020254 418179 867462 952821 985360 143012 759289 446224 887808 / 6 994423 > 494 [i]
- extracting embedded orthogonal array [i] would yield OA(494, 104, S4, 70), but