MinT - Best Known (220−109, 220, s)-Nets in Base 4

Best Known (220−109, 220, s)-Nets in Base 4

(220−109, 220, 130)-Net over F₄ — Constructive and digital

Digital (111, 220, 130)-net over F₄, using

t-expansion [i] based on digital (105, 220, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(220−109, 220, 165)-Net over F₄ — Digital

Digital (111, 220, 165)-net over F₄, using

t-expansion [i] based on digital (109, 220, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(220−109, 220, 1888)-Net in Base 4 — Upper bound on s

There is no (111, 220, 1889)-net in base 4, because

1 times m-reduction [i] would yield (111, 219, 1889)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 722970 355998 014217 494730 976065 900139 568986 537377 434329 588238 302566 102176 136321 046585 268287 151044 278597 513045 286967 885667 438419 199904 > 4²¹⁹ [i]