Best Known (165−125, 165, s)-Nets in Base 4
(165−125, 165, 56)-Net over F4 — Constructive and digital
Digital (40, 165, 56)-net over F4, using
- t-expansion [i] based on digital (33, 165, 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
(165−125, 165, 75)-Net over F4 — Digital
Digital (40, 165, 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
(165−125, 165, 169)-Net over F4 — Upper bound on s (digital)
There is no digital (40, 165, 170)-net over F4, because
- 1 times m-reduction [i] would yield digital (40, 164, 170)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4164, 170, F4, 124) (dual of [170, 6, 125]-code), but
- residual code [i] would yield linear OA(440, 45, F4, 31) (dual of [45, 5, 32]-code), but
- “Liz†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(440, 45, F4, 31) (dual of [45, 5, 32]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(4164, 170, F4, 124) (dual of [170, 6, 125]-code), but
(165−125, 165, 263)-Net in Base 4 — Upper bound on s
There is no (40, 165, 264)-net in base 4, because
- 1 times m-reduction [i] would yield (40, 164, 264)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 548 781902 976269 985410 270170 995035 023695 625385 473759 886862 256351 286072 682862 557341 328247 773844 498720 > 4164 [i]