Best Known (135−98, 135, s)-Nets in Base 4
(135−98, 135, 56)-Net over F4 — Constructive and digital
Digital (37, 135, 56)-net over F4, using
- t-expansion [i] based on digital (33, 135, 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
(135−98, 135, 66)-Net over F4 — Digital
Digital (37, 135, 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
(135−98, 135, 170)-Net over F4 — Upper bound on s (digital)
There is no digital (37, 135, 171)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4135, 171, F4, 98) (dual of [171, 36, 99]-code), but
- construction Y1 [i] would yield
- OA(4134, 149, S4, 98), but
- the linear programming bound shows that M ≥ 16 522880 710265 097728 979611 544086 920533 682517 483034 531558 847934 994306 996193 933029 168694 427648 / 26940 869703 > 4134 [i]
- OA(436, 171, S4, 22), but
- discarding factors would yield OA(436, 158, S4, 22), but
- the Rao or (dual) Hamming bound shows that M ≥ 4882 374384 904669 137046 > 436 [i]
- discarding factors would yield OA(436, 158, S4, 22), but
- OA(4134, 149, S4, 98), but
- construction Y1 [i] would yield
(135−98, 135, 252)-Net in Base 4 — Upper bound on s
There is no (37, 135, 253)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 2240 390062 880074 181404 542300 298707 874496 700247 513298 053192 489677 380354 137105 381864 > 4135 [i]