MinT - Best Known (227−123, 227, s)-Nets in Base 4

Best Known (227−123, 227, s)-Nets in Base 4

(227−123, 227, 104)-Net over F₄ — Constructive and digital

Digital (104, 227, 104)-net over F₄, using

t-expansion [i] based on digital (73, 227, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(227−123, 227, 144)-Net over F₄ — Digital

Digital (104, 227, 144)-net over F₄, using

t-expansion [i] based on digital (91, 227, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(227−123, 227, 1285)-Net in Base 4 — Upper bound on s

There is no (104, 227, 1286)-net in base 4, because

1 times m-reduction [i] would yield (104, 226, 1286)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11630 654183 317948 639953 048537 881858 529710 086965 482747 953014 623277 099477 359269 328754 878706 506101 211328 842913 069221 034058 287991 210509 187360 > 4²²⁶ [i]