MinT - Best Known (158−103, 158, s)-Nets in Base 4

Best Known (158−103, 158, s)-Nets in Base 4

(158−103, 158, 66)-Net over F₄ — Constructive and digital

Digital (55, 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]

(158−103, 158, 91)-Net over F₄ — Digital

Digital (55, 158, 91)-net over F₄, using

t-expansion [i] based on digital (50, 158, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(158−103, 158, 431)-Net in Base 4 — Upper bound on s

There is no (55, 158, 432)-net in base 4, because

1 times m-reduction [i] would yield (55, 157, 432)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34944 898995 572468 611015 842688 659589 340312 301239 372924 776193 856164 314255 838415 454209 752342 540776 > 4¹⁵⁷ [i]