MinT - Best Known (256−163, 256, s)-Nets in Base 4

Best Known (256−163, 256, s)-Nets in Base 4

(256−163, 256, 104)-Net over F₄ — Constructive and digital

Digital (93, 256, 104)-net over F₄, using

t-expansion [i] based on digital (73, 256, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(256−163, 256, 144)-Net over F₄ — Digital

Digital (93, 256, 144)-net over F₄, using

t-expansion [i] based on digital (91, 256, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(256−163, 256, 746)-Net in Base 4 — Upper bound on s

There is no (93, 256, 747)-net in base 4, because

1 times m-reduction [i] would yield (93, 255, 747)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3603 519266 354286 814381 711007 705630 184322 064169 550378 724443 686245 264691 407547 609722 448541 944222 723323 793515 989204 344498 052493 878750 301986 612874 214889 700452 > 4²⁵⁵ [i]