Best Known (25, ∞, s)-Nets in Base 4
(25, ∞, 34)-Net over F4 — Constructive and digital
Digital (25, m, 34)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (25, 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
(25, ∞, 35)-Net in Base 4 — Constructive
(25, m, 35)-net in base 4 for arbitrarily large m, using
- net from sequence [i] based on (25, 34)-sequence in base 4, using
- t-expansion [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
- t-expansion [i] based on (24, 34)-sequence in base 4, using
(25, ∞, 51)-Net over F4 — Digital
Digital (25, m, 51)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (25, 50)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 25 and N(F) ≥ 51, using
(25, ∞, 88)-Net in Base 4 — Upper bound on s
There is no (25, m, 89)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (25, 263, 89)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4263, 89, S4, 3, 238), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 67659 445738 247289 350861 614275 746211 549407 554318 212418 114991 831853 732713 615567 298049 719947 818359 296093 856232 445074 435336 813874 313725 207241 416729 622837 230003 290112 / 239 > 4263 [i]
- extracting embedded OOA [i] would yield OOA(4263, 89, S4, 3, 238), but