MinT - Best Known (188−129, 188, s)-Nets in Base 4

Best Known (188−129, 188, s)-Nets in Base 4

(188−129, 188, 66)-Net over F₄ — Constructive and digital

Digital (59, 188, 66)-net over F₄, using

t-expansion [i] based on digital (49, 188, 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]

(188−129, 188, 91)-Net over F₄ — Digital

Digital (59, 188, 91)-net over F₄, using

t-expansion [i] based on digital (50, 188, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(188−129, 188, 421)-Net in Base 4 — Upper bound on s

There is no (59, 188, 422)-net in base 4, because

1 times m-reduction [i] would yield (59, 187, 422)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40357 592171 141136 390139 921230 552933 574931 900593 930627 731394 184658 830816 670160 689084 548236 643225 397614 702916 889030 > 4¹⁸⁷ [i]