MinT - Best Known (189, 189+65, s)-Nets in Base 4

Best Known (189, 189+65, s)-Nets in Base 4

(189, 189+65, 548)-Net over F₄ — Constructive and digital

Digital (189, 254, 548)-net over F₄, using

4¹ times duplication [i] based on digital (188, 253, 548)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 37, 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 (151, 216, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 72, 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]

(189, 189+65, 648)-Net in Base 4 — Constructive

(189, 254, 648)-net in base 4, using

4² times duplication [i] based on (187, 252, 648)-net in base 4, using
- t-expansion [i] based on (185, 252, 648)-net in base 4, using
  - trace code for nets [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 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]

(189, 189+65, 2048)-Net over F₄ — Digital

Digital (189, 254, 2048)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(189, 189+65, 245329)-Net in Base 4 — Upper bound on s

There is no (189, 254, 245330)-net in base 4, because

1 times m-reduction [i] would yield (189, 253, 245330)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 209 513317 211499 625607 107717 463687 167514 959082 524318 219202 127216 165129 915985 859409 908534 703675 903386 144301 178529 671931 946444 169664 935163 896479 733755 939202 > 4²⁵³ [i]