Best Known (172−146, 172, s)-Nets in Base 4
(172−146, 172, 34)-Net over F4 — Constructive and digital
Digital (26, 172, 34)-net over F4, using
- t-expansion [i] based on digital (21, 172, 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
(172−146, 172, 36)-Net in Base 4 — Constructive
(26, 172, 36)-net in base 4, using
- net from sequence [i] based on (26, 35)-sequence in base 4, using
- base expansion [i] based on digital (52, 35)-sequence over F2, using
- t-expansion [i] based on digital (51, 35)-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 3 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]
- t-expansion [i] based on digital (51, 35)-sequence over F2, using
- base expansion [i] based on digital (52, 35)-sequence over F2, using
(172−146, 172, 55)-Net over F4 — Digital
Digital (26, 172, 55)-net over F4, using
- net from sequence [i] based on digital (26, 54)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 26 and N(F) ≥ 55, using
(172−146, 172, 111)-Net in Base 4 — Upper bound on s
There is no (26, 172, 112)-net in base 4, because
- 74 times m-reduction [i] would yield (26, 98, 112)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(498, 112, S4, 72), but
- the linear programming bound shows that M ≥ 1 166662 731140 735083 887873 150433 161508 032759 728734 208123 114163 798016 / 9 452405 > 498 [i]
- extracting embedded orthogonal array [i] would yield OA(498, 112, S4, 72), but