MinT - Best Known (250−180, 250, s)-Nets in Base 4

Best Known (250−180, 250, s)-Nets in Base 4

(250−180, 250, 66)-Net over F₄ — Constructive and digital

Digital (70, 250, 66)-net over F₄, using

t-expansion [i] based on digital (49, 250, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ 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]

(250−180, 250, 105)-Net over F₄ — Digital

Digital (70, 250, 105)-net over F₄, using

net from sequence [i] based on digital (70, 104)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
  - Drinfeld modules of rank 1 [i]

(250−180, 250, 393)-Net over F₄ — Upper bound on s (digital)

There is no digital (70, 250, 394)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁰, 394, F₄, 180) (dual of [394, 144, 181]-code), but
- residual code [i] would yield OA(4⁷⁰, 213, S₄, 45), but
  - the linear programming bound shows that MÂ â‰¥ 10138 160863 655691 762503 861603 731431 648994 530508 429756 680407 696662 337185 130982 211720 965164 892160 / 6819 841021 187779 545947 587667 623913 181856 482312 426343 > 4⁷⁰ [i]

(250−180, 250, 466)-Net in Base 4 — Upper bound on s

There is no (70, 250, 467)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 364632 974372 419588 296342 857677 057295 494285 238220 853084 044824 410749 480316 238841 217891 356519 997498 531498 386529 414283 727025 992399 976175 513231 858846 586064 > 4²⁵⁰ [i]