MinT - Best Known (138, 138+93, s)-Nets in Base 4

Best Known (138, 138+93, s)-Nets in Base 4

(138, 138+93, 137)-Net over F₄ — Constructive and digital

Digital (138, 231, 137)-net over F₄, using

7 times m-reduction [i] based on digital (138, 238, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 65, 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, 173, 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]

(138, 138+93, 325)-Net over F₄ — Digital

Digital (138, 231, 325)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+93, 6105)-Net in Base 4 — Upper bound on s

There is no (138, 231, 6106)-net in base 4, because

1 times m-reduction [i] would yield (138, 230, 6106)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 984918 476977 255669 411494 687578 113599 830283 084460 114527 783652 446419 860765 986129 882858 326197 362124 352296 136537 816950 912029 136495 460917 910576 > 4²³⁰ [i]