MinT - Best Known (227−147, 227, s)-Nets in Base 4

Best Known (227−147, 227, s)-Nets in Base 4

(227−147, 227, 104)-Net over F₄ — Constructive and digital

Digital (80, 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]

(227−147, 227, 112)-Net over F₄ — Digital

Digital (80, 227, 112)-net over F₄, using

t-expansion [i] based on digital (73, 227, 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]

(227−147, 227, 624)-Net in Base 4 — Upper bound on s

There is no (80, 227, 625)-net in base 4, because

1 times m-reduction [i] would yield (80, 226, 625)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12893 094420 957157 997501 232324 920256 447115 633693 330252 155679 811463 214494 047642 807466 787243 953816 606250 608807 520223 567467 245617 051168 233576 > 4²²⁶ [i]