MinT - Best Known (64, 64+103, s)-Nets in Base 4

Best Known (64, 64+103, s)-Nets in Base 4

(64, 64+103, 66)-Net over F₄ — Constructive and digital

Digital (64, 167, 66)-net over F₄, using

t-expansion [i] based on digital (49, 167, 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]

(64, 64+103, 99)-Net over F₄ — Digital

Digital (64, 167, 99)-net over F₄, using

t-expansion [i] based on digital (61, 167, 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]

(64, 64+103, 562)-Net in Base 4 — Upper bound on s

There is no (64, 167, 563)-net in base 4, because

1 times m-reduction [i] would yield (64, 166, 563)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9387 751099 027725 108086 657967 384931 398626 600344 051100 765343 646599 844624 310032 597456 886341 534790 924620 > 4¹⁶⁶ [i]