MinT - Best Known (260−113, 260, s)-Nets in Base 4

Best Known (260−113, 260, s)-Nets in Base 4

(260−113, 260, 137)-Net over F₄ — Constructive and digital

Digital (147, 260, 137)-net over F₄, using

t-expansion [i] based on digital (145, 260, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 72, 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, 188, 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]

(260−113, 260, 282)-Net over F₄ — Digital

Digital (147, 260, 282)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(260−113, 260, 4360)-Net in Base 4 — Upper bound on s

There is no (147, 260, 4361)-net in base 4, because

1 times m-reduction [i] would yield (147, 259, 4361)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 868634 820679 634686 997212 631781 636066 413290 944937 774950 816683 788013 981648 339117 613766 478805 952660 619645 868403 348986 043057 979192 993992 443730 294538 735759 148400 > 4²⁵⁹ [i]