MinT - Best Known (230−119, 230, s)-Nets in Base 4

Best Known (230−119, 230, s)-Nets in Base 4

(230−119, 230, 130)-Net over F₄ — Constructive and digital

Digital (111, 230, 130)-net over F₄, using

t-expansion [i] based on digital (105, 230, 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]

(230−119, 230, 165)-Net over F₄ — Digital

Digital (111, 230, 165)-net over F₄, using

t-expansion [i] based on digital (109, 230, 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]

(230−119, 230, 1603)-Net in Base 4 — Upper bound on s

There is no (111, 230, 1604)-net in base 4, because

1 times m-reduction [i] would yield (111, 229, 1604)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 745926 096346 838196 823044 824949 184378 609348 387005 744220 035846 132591 290652 382970 449561 550477 277627 196428 889009 720099 865715 505624 529477 532136 > 4²²⁹ [i]