Best Known (38, 144, s)-Nets in Base 4
(38, 144, 56)-Net over F4 — Constructive and digital
Digital (38, 144, 56)-net over F4, using
- t-expansion [i] based on digital (33, 144, 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
- net from sequence [i] based on digital (33, 55)-sequence over F4, using
(38, 144, 66)-Net over F4 — Digital
Digital (38, 144, 66)-net over F4, using
- t-expansion [i] based on digital (37, 144, 66)-net over F4, using
- net from sequence [i] based on digital (37, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 37 and N(F) ≥ 66, using
- net from sequence [i] based on digital (37, 65)-sequence over F4, using
(38, 144, 198)-Net over F4 — Upper bound on s (digital)
There is no digital (38, 144, 199)-net over F4, because
- 2 times m-reduction [i] would yield digital (38, 142, 199)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4142, 199, F4, 104) (dual of [199, 57, 105]-code), but
- construction Y1 [i] would yield
- linear OA(4141, 163, F4, 104) (dual of [163, 22, 105]-code), but
- construction Y1 [i] would yield
- OA(4140, 151, S4, 104), but
- 2 times truncation [i] would yield OA(4138, 149, S4, 102), but
- the linear programming bound shows that M ≥ 11 769278 720446 703333 267659 747956 551886 187319 922656 933070 157940 129170 842786 675855 465333 653504 / 85 907459 > 4138 [i]
- 2 times truncation [i] would yield OA(4138, 149, S4, 102), but
- OA(422, 163, S4, 12), but
- the linear programming bound shows that M ≥ 22822 886502 605724 057600 / 1273 153051 > 422 [i]
- OA(4140, 151, S4, 104), but
- construction Y1 [i] would yield
- OA(457, 199, S4, 36), but
- discarding factors would yield OA(457, 195, S4, 36), but
- the linear programming bound shows that M ≥ 494 490067 436667 455875 598203 818411 591542 653847 684514 372082 636620 702806 966272 / 22309 972171 916724 959618 830756 395146 773299 > 457 [i]
- discarding factors would yield OA(457, 195, S4, 36), but
- linear OA(4141, 163, F4, 104) (dual of [163, 22, 105]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4142, 199, F4, 104) (dual of [199, 57, 105]-code), but
(38, 144, 255)-Net in Base 4 — Upper bound on s
There is no (38, 144, 256)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 552 724342 739735 384585 163583 461979 462331 793806 695022 910459 059681 191039 433394 660082 426969 > 4144 [i]