MinT - Best Known (211−57, 211, s)-Nets in Base 4

Best Known (211−57, 211, s)-Nets in Base 4

(211−57, 211, 531)-Net over F₄ — Constructive and digital

Digital (154, 211, 531)-net over F₄, using

t-expansion [i] based on digital (153, 211, 531)-net over F₄, using
- 8 times m-reduction [i] based on 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]

(211−57, 211, 576)-Net in Base 4 — Constructive

(154, 211, 576)-net in base 4, using

4¹ times duplication [i] based on (153, 210, 576)-net in base 4, using
- trace code for nets [i] based on (13, 70, 192)-net in base 64, using
  - base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(211−57, 211, 1371)-Net over F₄ — Digital

Digital (154, 211, 1371)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−57, 211, 123382)-Net in Base 4 — Upper bound on s

There is no (154, 211, 123383)-net in base 4, because

1 times m-reduction [i] would yield (154, 210, 123383)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 708029 827878 652385 488928 962009 245083 485238 240283 960124 787475 436658 065695 901856 485340 127186 160652 972233 356193 145482 103956 987940 > 4²¹⁰ [i]