MinT - Best Known (111, 111+127, s)-Nets in Base 4

Best Known (111, 111+127, s)-Nets in Base 4

(111, 111+127, 130)-Net over F₄ — Constructive and digital

Digital (111, 238, 130)-net over F₄, using

t-expansion [i] based on digital (105, 238, 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]

(111, 111+127, 165)-Net over F₄ — Digital

Digital (111, 238, 165)-net over F₄, using

t-expansion [i] based on digital (109, 238, 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]

(111, 111+127, 1439)-Net in Base 4 — Upper bound on s

There is no (111, 238, 1440)-net in base 4, because

1 times m-reduction [i] would yield (111, 237, 1440)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49700 792823 451089 290188 928790 376594 852170 704452 532847 258347 527225 657825 131414 207757 350167 515594 087007 176355 206255 692823 517519 507203 700390 143204 > 4²³⁷ [i]