MinT - Best Known (66, 66+89, s)-Nets in Base 4

Best Known (66, 66+89, s)-Nets in Base 4

(66, 66+89, 66)-Net over F₄ — Constructive and digital

Digital (66, 155, 66)-net over F₄, using

t-expansion [i] based on digital (49, 155, 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]

(66, 66+89, 99)-Net over F₄ — Digital

Digital (66, 155, 99)-net over F₄, using

t-expansion [i] based on digital (61, 155, 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]

(66, 66+89, 700)-Net in Base 4 — Upper bound on s

There is no (66, 155, 701)-net in base 4, because

1 times m-reduction [i] would yield (66, 154, 701)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 530 188896 216418 087435 874322 671331 309911 982799 103832 059163 698746 164944 077687 157891 181854 138137 > 4¹⁵⁴ [i]