Best Known (139−102, 139, s)-Nets in Base 4
(139−102, 139, 56)-Net over F4 — Constructive and digital
Digital (37, 139, 56)-net over F4, using
- t-expansion [i] based on digital (33, 139, 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
(139−102, 139, 66)-Net over F4 — Digital
Digital (37, 139, 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
(139−102, 139, 162)-Net over F4 — Upper bound on s (digital)
There is no digital (37, 139, 163)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4139, 163, F4, 102) (dual of [163, 24, 103]-code), but
- construction Y1 [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]
- OA(424, 163, S4, 14), but
- discarding factors would yield OA(424, 134, S4, 14), but
- the Rao or (dual) Hamming bound shows that M ≥ 292 368491 029312 > 424 [i]
- discarding factors would yield OA(424, 134, S4, 14), but
- OA(4138, 149, S4, 102), but
- construction Y1 [i] would yield
(139−102, 139, 249)-Net in Base 4 — Upper bound on s
There is no (37, 139, 250)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 515296 419133 893906 436332 865684 443113 744987 969688 903379 417021 364648 956609 856248 812556 > 4139 [i]