MinT - Best Known (157−93, 157, s)-Nets in Base 4

Best Known (157−93, 157, s)-Nets in Base 4

(157−93, 157, 66)-Net over F₄ — Constructive and digital

Digital (64, 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]

(157−93, 157, 99)-Net over F₄ — Digital

Digital (64, 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]

(157−93, 157, 623)-Net in Base 4 — Upper bound on s

There is no (64, 157, 624)-net in base 4, because

1 times m-reduction [i] would yield (64, 156, 624)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8652 423017 784279 412186 669440 129941 770636 799833 956664 395970 499609 984881 478921 809789 059794 546713 > 4¹⁵⁶ [i]