MinT - Best Known (186−119, 186, s)-Nets in Base 4

Best Known (186−119, 186, s)-Nets in Base 4

(186−119, 186, 66)-Net over F₄ — Constructive and digital

Digital (67, 186, 66)-net over F₄, using

t-expansion [i] based on digital (49, 186, 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]

(186−119, 186, 99)-Net over F₄ — Digital

Digital (67, 186, 99)-net over F₄, using

t-expansion [i] based on digital (61, 186, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(186−119, 186, 540)-Net in Base 4 — Upper bound on s

There is no (67, 186, 541)-net in base 4, because

1 times m-reduction [i] would yield (67, 185, 541)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2568 917681 905544 140172 349775 069907 233965 138097 329753 515774 224772 053889 817831 210045 241949 190610 479669 890747 316160 > 4¹⁸⁵ [i]