MinT - Best Known (158−97, 158, s)-Nets in Base 4

Best Known (158−97, 158, s)-Nets in Base 4

(158−97, 158, 66)-Net over F₄ — Constructive and digital

Digital (61, 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]

(158−97, 158, 99)-Net over F₄ — Digital

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]

(158−97, 158, 543)-Net in Base 4 — Upper bound on s

There is no (61, 158, 544)-net in base 4, because

1 times m-reduction [i] would yield (61, 157, 544)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34872 053426 597788 103115 545567 866081 573952 144160 649397 722825 080322 335483 645050 400513 270380 809456 > 4¹⁵⁷ [i]