MinT - Best Known (258−66, 258, s)-Nets in Base 4

Best Known (258−66, 258, s)-Nets in Base 4

(258−66, 258, 548)-Net over F₄ — Constructive and digital

Digital (192, 258, 548)-net over F₄, using

4¹ times duplication [i] based on digital (191, 257, 548)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 38, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (153, 219, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 73, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(258−66, 258, 648)-Net in Base 4 — Constructive

(192, 258, 648)-net in base 4, using

t-expansion [i] based on (190, 258, 648)-net in base 4, using
- trace code for nets [i] based on (18, 86, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
    - base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(258−66, 258, 2080)-Net over F₄ — Digital

Digital (192, 258, 2080)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−66, 258, 223457)-Net in Base 4 — Upper bound on s

There is no (192, 258, 223458)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 214541 914772 341423 736705 884741 652372 573905 064537 625778 364139 088406 441066 433214 002742 790215 780945 739556 375295 923067 008967 338509 926393 954079 190868 041061 380445 > 4²⁵⁸ [i]