Best Known (111−78, 111, s)-Nets in Base 4
(111−78, 111, 56)-Net over F4 — Constructive and digital
Digital (33, 111, 56)-net over F4, using
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 56, using
- F5 from the 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) = 33 and N(F) ≥ 56, using
(111−78, 111, 65)-Net over F4 — Digital
Digital (33, 111, 65)-net over F4, using
- net from sequence [i] based on digital (33, 64)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 33 and N(F) ≥ 65, using
(111−78, 111, 225)-Net over F4 — Upper bound on s (digital)
There is no digital (33, 111, 226)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4111, 226, F4, 78) (dual of [226, 115, 79]-code), but
- construction Y1 [i] would yield
- OA(4110, 145, S4, 78), but
- the linear programming bound shows that M ≥ 49373 306964 861862 839083 768214 951308 945061 380326 149716 021338 126814 314674 695519 115905 973031 534592 / 23434 367822 584829 194407 918903 > 4110 [i]
- linear OA(4115, 226, F4, 81) (dual of [226, 111, 82]-code), but
- discarding factors / shortening the dual code would yield linear OA(4115, 219, F4, 81) (dual of [219, 104, 82]-code), but
- construction Y1 [i] would yield
- OA(4114, 146, S4, 81), but
- the linear programming bound shows that M ≥ 7 594596 073360 304072 815867 736592 315637 386549 140598 576996 791410 829396 630589 236390 143121 686528 / 15336 256272 900153 260065 > 4114 [i]
- OA(4104, 219, S4, 73), but
- discarding factors would yield OA(4104, 150, S4, 73), but
- the linear programming bound shows that M ≥ 32 028783 264027 932827 838658 827950 132017 443030 031565 779091 184319 519613 927802 631343 679332 066747 830675 439616 / 70932 376441 373736 310653 589982 582096 950125 > 4104 [i]
- discarding factors would yield OA(4104, 150, S4, 73), but
- OA(4114, 146, S4, 81), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(4115, 219, F4, 81) (dual of [219, 104, 82]-code), but
- OA(4110, 145, S4, 78), but
- construction Y1 [i] would yield
(111−78, 111, 234)-Net in Base 4 — Upper bound on s
There is no (33, 111, 235)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 6 974965 211470 815411 274289 810557 902542 856926 295099 006724 017481 041760 > 4111 [i]