MinT - Best Known (111, 111+115, s)-Nets in Base 4

Best Known (111, 111+115, s)-Nets in Base 4

(111, 111+115, 130)-Net over F₄ — Constructive and digital

Digital (111, 226, 130)-net over F₄, using

t-expansion [i] based on digital (105, 226, 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+115, 165)-Net over F₄ — Digital

Digital (111, 226, 165)-net over F₄, using

t-expansion [i] based on digital (109, 226, 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+115, 1705)-Net in Base 4 — Upper bound on s

There is no (111, 226, 1706)-net in base 4, because

1 times m-reduction [i] would yield (111, 225, 1706)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2999 350920 970499 351729 601578 605698 867846 489196 594932 596160 952877 720527 970056 095265 566036 531420 764092 906184 734552 864911 634190 360111 394760 > 4²²⁵ [i]