MinT - Best Known (111, 111+113, s)-Nets in Base 4

Best Known (111, 111+113, s)-Nets in Base 4

(111, 111+113, 130)-Net over F₄ — Constructive and digital

Digital (111, 224, 130)-net over F₄, using

t-expansion [i] based on digital (105, 224, 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+113, 165)-Net over F₄ — Digital

Digital (111, 224, 165)-net over F₄, using

t-expansion [i] based on digital (109, 224, 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+113, 1761)-Net in Base 4 — Upper bound on s

There is no (111, 224, 1762)-net in base 4, because

1 times m-reduction [i] would yield (111, 223, 1762)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 184 625824 600389 654889 107402 044438 627269 577089 422094 133314 493169 943637 997650 015310 001542 124921 563929 892818 657817 825036 482483 525001 127960 > 4²²³ [i]