MinT - Best Known (140, 140+101, s)-Nets in Base 4

Best Known (140, 140+101, s)-Nets in Base 4

(140, 140+101, 137)-Net over F₄ — Constructive and digital

Digital (140, 241, 137)-net over F₄, using

3 times m-reduction [i] based on digital (140, 244, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 67, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 177, 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]

(140, 140+101, 297)-Net over F₄ — Digital

Digital (140, 241, 297)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(140, 140+101, 4998)-Net in Base 4 — Upper bound on s

There is no (140, 241, 4999)-net in base 4, because

1 times m-reduction [i] would yield (140, 240, 4999)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 123782 433949 746061 486528 272973 482939 807197 122476 538524 555391 434832 864723 524116 483200 002582 778799 761210 138730 883487 939133 363660 001516 491568 217576 > 4²⁴⁰ [i]