MinT - Best Known (227−113, 227, s)-Nets in Base 4

Best Known (227−113, 227, s)-Nets in Base 4

(227−113, 227, 130)-Net over F₄ — Constructive and digital

Digital (114, 227, 130)-net over F₄, using

t-expansion [i] based on digital (105, 227, 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]

(227−113, 227, 165)-Net over F₄ — Digital

Digital (114, 227, 165)-net over F₄, using

t-expansion [i] based on digital (109, 227, 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]

(227−113, 227, 1900)-Net in Base 4 — Upper bound on s

There is no (114, 227, 1901)-net in base 4, because

1 times m-reduction [i] would yield (114, 226, 1901)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11703 959284 293605 513268 753247 610706 111391 083875 615912 475366 170759 598214 277294 048386 546199 369705 397749 934636 931721 149287 183399 302722 044220 > 4²²⁶ [i]