Best Known (135−105, 135, s)-Nets in Base 5
(135−105, 135, 51)-Net over F5 — Constructive and digital
Digital (30, 135, 51)-net over F5, using
- t-expansion [i] based on digital (22, 135, 51)-net over F5, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 22 and N(F) ≥ 51, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
(135−105, 135, 58)-Net over F5 — Digital
Digital (30, 135, 58)-net over F5, using
- net from sequence [i] based on digital (30, 57)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 30 and N(F) ≥ 58, using
(135−105, 135, 170)-Net over F5 — Upper bound on s (digital)
There is no digital (30, 135, 171)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(5135, 171, F5, 105) (dual of [171, 36, 106]-code), but
- construction Y1 [i] would yield
- OA(5134, 147, S5, 105), but
- the linear programming bound shows that M ≥ 35 088836 405859 599893 783329 592875 489923 934405 996939 363274 030802 657132 976918 319400 283508 002758 026123 046875 / 7350 907663 > 5134 [i]
- OA(536, 171, S5, 24), but
- the Rao or (dual) Hamming bound shows that M ≥ 15 032494 166705 850791 777885 > 536 [i]
- OA(5134, 147, S5, 105), but
- construction Y1 [i] would yield
(135−105, 135, 282)-Net in Base 5 — Upper bound on s
There is no (30, 135, 283)-net in base 5, because
- 1 times m-reduction [i] would yield (30, 134, 283)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 4616 502925 285325 209856 195033 511914 284712 286975 364951 815320 111862 389519 360729 709821 459817 616305 > 5134 [i]