MinT - Best Known (252−67, 252, s)-Nets in Base 4

Best Known (252−67, 252, s)-Nets in Base 4

(252−67, 252, 531)-Net over F₄ — Constructive and digital

Digital (185, 252, 531)-net over F₄, using

t-expansion [i] based on digital (179, 252, 531)-net over F₄, using
- 6 times m-reduction [i] based on digital (179, 258, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 86, 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]

(252−67, 252, 648)-Net in Base 4 — Constructive

(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]

(252−67, 252, 1715)-Net over F₄ — Digital

Digital (185, 252, 1715)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−67, 252, 166520)-Net in Base 4 — Upper bound on s

There is no (185, 252, 166521)-net in base 4, because

1 times m-reduction [i] would yield (185, 251, 166521)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 095331 413252 127905 511046 108848 255483 470388 631083 030005 317011 480361 468234 494357 387915 714590 104628 249214 907918 653654 490328 825949 474090 079229 860057 061916 > 4²⁵¹ [i]