Best Known (26, 26+46, s)-Nets in Base 4
(26, 26+46, 34)-Net over F4 — Constructive and digital
Digital (26, 72, 34)-net over F4, using
- t-expansion [i] based on digital (21, 72, 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
(26, 26+46, 36)-Net in Base 4 — Constructive
(26, 72, 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
(26, 26+46, 55)-Net over F4 — Digital
Digital (26, 72, 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
(26, 26+46, 222)-Net in Base 4 — Upper bound on s
There is no (26, 72, 223)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 22 546582 773050 935969 355454 324369 997221 414340 > 472 [i]
- extracting embedded orthogonal array [i] would yield OA(472, 223, S4, 46), but
- the linear programming bound shows that M ≥ 360 417143 929919 650649 839526 859205 286331 400168 894487 479850 253672 313831 388847 004885 511580 359694 158397 440000 000000 / 15 992885 341550 871668 440406 576991 261654 953271 872698 021682 171330 664199 > 472 [i]