Best Known (42, 42+136, s)-Nets in Base 4
(42, 42+136, 56)-Net over F4 — Constructive and digital
Digital (42, 178, 56)-net over F4, using
- t-expansion [i] based on digital (33, 178, 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
(42, 42+136, 75)-Net over F4 — Digital
Digital (42, 178, 75)-net over F4, using
- t-expansion [i] based on digital (40, 178, 75)-net over F4, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 40 and N(F) ≥ 75, using
- net from sequence [i] based on digital (40, 74)-sequence over F4, using
(42, 42+136, 175)-Net over F4 — Upper bound on s (digital)
There is no digital (42, 178, 176)-net over F4, because
- 8 times m-reduction [i] would yield digital (42, 170, 176)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4170, 176, F4, 128) (dual of [176, 6, 129]-code), but
- construction Y1 [i] would yield
- linear OA(4169, 173, F4, 128) (dual of [173, 4, 129]-code), but
- linear OA(46, 176, F4, 3) (dual of [176, 170, 4]-code or 176-cap in PG(5,4)), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(4170, 176, F4, 128) (dual of [176, 6, 129]-code), but
(42, 42+136, 275)-Net in Base 4 — Upper bound on s
There is no (42, 178, 276)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 158823 331900 126686 509910 123952 610682 466064 334408 323453 731174 363602 337060 749407 945445 960141 001635 679242 199600 > 4178 [i]