Best Known (32, ∞, s)-Nets in Base 4
(32, ∞, 34)-Net over F4 — Constructive and digital
Digital (32, m, 34)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (32, 33)-sequence over F4, using
- t-expansion [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
- t-expansion [i] based on digital (21, 33)-sequence over F4, using
(32, ∞, 43)-Net in Base 4 — Constructive
(32, m, 43)-net in base 4 for arbitrarily large m, using
- net from sequence [i] based on (32, 42)-sequence in base 4, using
- t-expansion [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
- t-expansion [i] based on (30, 42)-sequence in base 4, using
(32, ∞, 60)-Net over F4 — Digital
Digital (32, m, 60)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (32, 59)-sequence over F4, using
- t-expansion [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
- t-expansion [i] based on digital (31, 59)-sequence over F4, using
(32, ∞, 109)-Net in Base 4 — Upper bound on s
There is no (32, m, 110)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (32, 435, 110)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4435, 110, S4, 4, 403), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 054875 063481 616124 677859 711331 326146 029316 510093 812405 543725 714230 154498 556273 149990 616921 915800 320772 623552 815563 309832 963862 612446 544498 079601 274639 873255 874571 437476 078886 965066 542754 190312 342423 904086 653876 992402 014005 611975 088780 301577 371103 880548 449901 346816 / 101 > 4435 [i]
- extracting embedded OOA [i] would yield OOA(4435, 110, S4, 4, 403), but