Best Known (42, 42+134, s)-Nets in Base 4
(42, 42+134, 56)-Net over F4 — Constructive and digital
Digital (42, 176, 56)-net over F4, using
- t-expansion [i] based on digital (33, 176, 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+134, 75)-Net over F4 — Digital
Digital (42, 176, 75)-net over F4, using
- t-expansion [i] based on digital (40, 176, 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+134, 175)-Net over F4 — Upper bound on s (digital)
There is no digital (42, 176, 176)-net over F4, because
- 6 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+134, 275)-Net in Base 4 — Upper bound on s
There is no (42, 176, 276)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 9405 194934 670616 890309 915476 875183 251961 405934 384420 986330 736566 405851 266155 994143 570223 843860 118970 242100 > 4176 [i]