Best Known (31, 31+77, s)-Nets in Base 4
(31, 31+77, 34)-Net over F4 — Constructive and digital
Digital (31, 108, 34)-net over F4, using
- t-expansion [i] based on digital (21, 108, 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
(31, 31+77, 43)-Net in Base 4 — Constructive
(31, 108, 43)-net in base 4, using
- t-expansion [i] based on (30, 108, 43)-net in base 4, using
- net from sequence [i] based on (30, 42)-sequence in base 4, using
- base expansion [i] based on digital (60, 42)-sequence over F2, using
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- t-expansion [i] based on digital (59, 42)-sequence over F2, using
- base expansion [i] based on digital (60, 42)-sequence over F2, using
- net from sequence [i] based on (30, 42)-sequence in base 4, using
(31, 31+77, 60)-Net over F4 — Digital
Digital (31, 108, 60)-net over F4, using
- net from sequence [i] based on digital (31, 59)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 31 and N(F) ≥ 60, using
(31, 31+77, 136)-Net in Base 4 — Upper bound on s
There is no (31, 108, 137)-net in base 4, because
- extracting embedded orthogonal array [i] would yield OA(4108, 137, S4, 77), but
- the linear programming bound shows that M ≥ 217 390859 331981 319138 219553 931378 186456 570350 958702 486339 017246 642987 932425 092568 449024 / 1975 301566 404582 353457 > 4108 [i]