Best Known (40, 40+128, s)-Nets in Base 4
(40, 40+128, 56)-Net over F4 — Constructive and digital
Digital (40, 168, 56)-net over F4, using
- t-expansion [i] based on digital (33, 168, 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
(40, 40+128, 75)-Net over F4 — Digital
Digital (40, 168, 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
(40, 40+128, 169)-Net over F4 — Upper bound on s (digital)
There is no digital (40, 168, 170)-net over F4, because
- 4 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
(40, 40+128, 263)-Net in Base 4 — Upper bound on s
There is no (40, 168, 264)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 162039 652719 005475 927663 590208 589082 047752 888334 214691 857214 192672 275810 784983 533993 018382 401598 218972 > 4168 [i]