Best Known (251−190, 251, s)-Nets in Base 4
(251−190, 251, 66)-Net over F4 — Constructive and digital
Digital (61, 251, 66)-net over F4, using
- t-expansion [i] based on digital (49, 251, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second 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) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(251−190, 251, 99)-Net over F4 — Digital
Digital (61, 251, 99)-net over F4, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 61 and N(F) ≥ 99, using
(251−190, 251, 254)-Net over F4 — Upper bound on s (digital)
There is no digital (61, 251, 255)-net over F4, because
- 6 times m-reduction [i] would yield digital (61, 245, 255)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4245, 255, F4, 184) (dual of [255, 10, 185]-code), but
- residual code [i] would yield linear OA(461, 70, F4, 46) (dual of [70, 9, 47]-code), but
- “Gur†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(461, 70, F4, 46) (dual of [70, 9, 47]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(4245, 255, F4, 184) (dual of [255, 10, 185]-code), but
(251−190, 251, 395)-Net in Base 4 — Upper bound on s
There is no (61, 251, 396)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 14 645410 009058 136314 534889 384620 523960 247526 703463 295726 149042 906341 080414 257468 417117 420721 619587 715842 035513 942213 410684 246018 163636 487755 732268 082784 > 4251 [i]