MinT - Best Known (66, 66+91, s)-Nets in Base 4

Best Known (66, 66+91, s)-Nets in Base 4

(66, 66+91, 66)-Net over F₄ — Constructive and digital

Digital (66, 157, 66)-net over F₄, using

t-expansion [i] based on digital (49, 157, 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+91, 99)-Net over F₄ — Digital

Digital (66, 157, 99)-net over F₄, using

t-expansion [i] based on digital (61, 157, 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+91, 681)-Net in Base 4 — Upper bound on s

There is no (66, 157, 682)-net in base 4, because

1 times m-reduction [i] would yield (66, 156, 682)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8435 707244 869418 537032 140753 350725 755857 649785 752092 946241 577276 996897 708165 892051 053709 644640 > 4¹⁵⁶ [i]