MinT - Best Known (181−111, 181, s)-Nets in Base 4

Best Known (181−111, 181, s)-Nets in Base 4

(181−111, 181, 66)-Net over F₄ — Constructive and digital

Digital (70, 181, 66)-net over F₄, using

t-expansion [i] based on digital (49, 181, 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]

(181−111, 181, 105)-Net over F₄ — Digital

Digital (70, 181, 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]

(181−111, 181, 620)-Net in Base 4 — Upper bound on s

There is no (70, 181, 621)-net in base 4, because

1 times m-reduction [i] would yield (70, 180, 621)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 512515 953348 880876 808532 981289 226267 445012 155302 403308 750249 819568 290997 562174 433241 332615 479015 068023 041056 > 4¹⁸⁰ [i]