MinT - Best Known (64, 64+89, s)-Nets in Base 4

Best Known (64, 64+89, s)-Nets in Base 4

(64, 64+89, 66)-Net over F₄ — Constructive and digital

Digital (64, 153, 66)-net over F₄, using

t-expansion [i] based on digital (49, 153, 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+89, 99)-Net over F₄ — Digital

Digital (64, 153, 99)-net over F₄, using

t-expansion [i] based on digital (61, 153, 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+89, 655)-Net in Base 4 — Upper bound on s

There is no (64, 153, 656)-net in base 4, because

1 times m-reduction [i] would yield (64, 152, 656)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 33 001203 857160 253123 420168 725836 930199 541619 092477 908050 120483 367857 893673 316524 055827 112196 > 4¹⁵² [i]