MinT - Best Known (231−131, 231, s)-Nets in Base 4

Best Known (231−131, 231, s)-Nets in Base 4

(231−131, 231, 104)-Net over F₄ — Constructive and digital

Digital (100, 231, 104)-net over F₄, using

t-expansion [i] based on digital (73, 231, 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]

(231−131, 231, 144)-Net over F₄ — Digital

Digital (100, 231, 144)-net over F₄, using

t-expansion [i] based on digital (91, 231, 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]

(231−131, 231, 1074)-Net in Base 4 — Upper bound on s

There is no (100, 231, 1075)-net in base 4, because

1 times m-reduction [i] would yield (100, 230, 1075)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 086839 040891 927244 550974 254788 074016 303659 568031 624669 333177 705533 055416 102322 779114 215311 800680 166934 898304 497669 122063 044146 742453 621408 > 4²³⁰ [i]