MinT - Best Known (205−81, 205, s)-Nets in Base 4

Best Known (205−81, 205, s)-Nets in Base 4

(205−81, 205, 131)-Net over F₄ — Constructive and digital

Digital (124, 205, 131)-net over F₄, using

1 times m-reduction [i] based on digital (124, 206, 131)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 51, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 155, 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]

(205−81, 205, 311)-Net over F₄ — Digital

Digital (124, 205, 311)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(205−81, 205, 6149)-Net in Base 4 — Upper bound on s

There is no (124, 205, 6150)-net in base 4, because

1 times m-reduction [i] would yield (124, 204, 6150)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 661 484125 139952 927902 127846 353788 195598 204410 301386 538866 270475 688287 559901 976721 286066 724883 938337 524266 206815 629937 238784 > 4²⁰⁴ [i]