MinT - Best Known (212−54, 212, s)-Nets in Base 4

Best Known (212−54, 212, s)-Nets in Base 4

(212−54, 212, 541)-Net over F₄ — Constructive and digital

Digital (158, 212, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 29, 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 (129, 183, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 61, 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]

(212−54, 212, 648)-Net in Base 4 — Constructive

(158, 212, 648)-net in base 4, using

4² times duplication [i] based on (156, 210, 648)-net in base 4, using
- t-expansion [i] based on (155, 210, 648)-net in base 4, using
  - trace code for nets [i] based on (15, 70, 216)-net in base 64, using
    - base change [i] based on digital (5, 60, 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]

(212−54, 212, 1784)-Net over F₄ — Digital

Digital (158, 212, 1784)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−54, 212, 194322)-Net in Base 4 — Upper bound on s

There is no (158, 212, 194323)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 328341 722002 831153 642442 701850 730572 460332 268653 953217 039127 483107 475141 221785 792906 497735 222495 709398 142139 657134 409675 253120 > 4²¹² [i]