MinT - Best Known (243−139, 243, s)-Nets in Base 4

Best Known (243−139, 243, s)-Nets in Base 4

(243−139, 243, 104)-Net over F₄ — Constructive and digital

Digital (104, 243, 104)-net over F₄, using

t-expansion [i] based on digital (73, 243, 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]

(243−139, 243, 144)-Net over F₄ — Digital

Digital (104, 243, 144)-net over F₄, using

t-expansion [i] based on digital (91, 243, 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]

(243−139, 243, 1087)-Net in Base 4 — Upper bound on s

There is no (104, 243, 1088)-net in base 4, because

1 times m-reduction [i] would yield (104, 242, 1088)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 236508 667975 236848 994157 975467 041652 503654 844694 506399 657781 978921 594684 831096 901676 310122 342781 883463 364254 648069 328943 494459 290848 489601 653520 > 4²⁴² [i]