MinT - Best Known (231−151, 231, s)-Nets in Base 4

Best Known (231−151, 231, s)-Nets in Base 4

(231−151, 231, 104)-Net over F₄ — Constructive and digital

Digital (80, 231, 104)-net over F₄, using

t-expansion [i] based on digital (73, 231, 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]

(231−151, 231, 112)-Net over F₄ — Digital

Digital (80, 231, 112)-net over F₄, using

t-expansion [i] based on digital (73, 231, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(231−151, 231, 612)-Net in Base 4 — Upper bound on s

There is no (80, 231, 613)-net in base 4, because

1 times m-reduction [i] would yield (80, 230, 613)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 067348 526288 282850 564183 154293 113917 003545 186377 703616 230879 629004 342433 956573 025911 529216 223643 473059 390834 182584 364388 914386 828904 243856 > 4²³⁰ [i]