MinT - Best Known (158−95, 158, s)-Nets in Base 4

Best Known (158−95, 158, s)-Nets in Base 4

(158−95, 158, 66)-Net over F₄ — Constructive and digital

Digital (63, 158, 66)-net over F₄, using

t-expansion [i] based on digital (49, 158, 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]

(158−95, 158, 99)-Net over F₄ — Digital

Digital (63, 158, 99)-net over F₄, using

t-expansion [i] based on digital (61, 158, 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]

(158−95, 158, 590)-Net in Base 4 — Upper bound on s

There is no (63, 158, 591)-net in base 4, because

1 times m-reduction [i] would yield (63, 157, 591)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34804 281995 782067 916093 447155 370635 908835 528729 142945 173135 171615 797808 215400 386972 664445 482632 > 4¹⁵⁷ [i]