MinT - Best Known (165−107, 165, s)-Nets in Base 4

Best Known (165−107, 165, s)-Nets in Base 4

(165−107, 165, 66)-Net over F₄ — Constructive and digital

Digital (58, 165, 66)-net over F₄, using

t-expansion [i] based on digital (49, 165, 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]

(165−107, 165, 91)-Net over F₄ — Digital

Digital (58, 165, 91)-net over F₄, using

t-expansion [i] based on digital (50, 165, 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]

(165−107, 165, 458)-Net in Base 4 — Upper bound on s

There is no (58, 165, 459)-net in base 4, because

1 times m-reduction [i] would yield (58, 164, 459)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 576 163005 906264 301518 060898 694009 070961 362892 285917 230690 275702 952249 064652 358847 424024 966925 214640 > 4¹⁶⁴ [i]