MinT - Best Known (257−67, 257, s)-Nets in Base 4

Best Known (257−67, 257, s)-Nets in Base 4

(257−67, 257, 541)-Net over F₄ — Constructive and digital

Digital (190, 257, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 35, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (155, 222, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 74, 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]

(257−67, 257, 648)-Net in Base 4 — Constructive

(190, 257, 648)-net in base 4, using

1 times m-reduction [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]

(257−67, 257, 1905)-Net over F₄ — Digital

Digital (190, 257, 1905)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−67, 257, 205447)-Net in Base 4 — Upper bound on s

There is no (190, 257, 205448)-net in base 4, because

1 times m-reduction [i] would yield (190, 256, 205448)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13407 946386 209734 691929 663277 249099 935131 120306 026858 266680 536486 294607 116562 044042 568671 074178 485982 637852 569252 683900 924466 548418 864994 033560 622625 276300 > 4²⁵⁶ [i]