MinT - Best Known (166−99, 166, s)-Nets in Base 4

Best Known (166−99, 166, s)-Nets in Base 4

(166−99, 166, 66)-Net over F₄ — Constructive and digital

Digital (67, 166, 66)-net over F₄, using

t-expansion [i] based on digital (49, 166, 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]

(166−99, 166, 99)-Net over F₄ — Digital

Digital (67, 166, 99)-net over F₄, using

t-expansion [i] based on digital (61, 166, 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]

(166−99, 166, 639)-Net in Base 4 — Upper bound on s

There is no (67, 166, 640)-net in base 4, because

1 times m-reduction [i] would yield (67, 165, 640)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2346 356357 803442 800369 183032 344640 927080 938459 285126 284319 646842 435241 018596 507791 867832 815840 551869 > 4¹⁶⁵ [i]