MinT - Best Known (256−69, 256, s)-Nets in Base 4

Best Known (256−69, 256, s)-Nets in Base 4

(256−69, 256, 531)-Net over F₄ — Constructive and digital

Digital (187, 256, 531)-net over F₄, using

t-expansion [i] based on digital (179, 256, 531)-net over F₄, using
- 2 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]

(256−69, 256, 576)-Net in Base 4 — Constructive

(187, 256, 576)-net in base 4, using

4⁴ times duplication [i] based on (183, 252, 576)-net in base 4, using
- trace code for nets [i] based on (15, 84, 192)-net in base 64, using
  - base change [i] based on digital (3, 72, 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]

(256−69, 256, 1631)-Net over F₄ — Digital

Digital (187, 256, 1631)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−69, 256, 147816)-Net in Base 4 — Upper bound on s

There is no (187, 256, 147817)-net in base 4, because

1 times m-reduction [i] would yield (187, 255, 147817)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3352 486399 792839 834120 573247 522029 452846 226444 848195 744010 405637 263950 830468 822905 512582 056327 974846 985900 987577 759256 352600 438426 533921 852174 572020 153060 > 4²⁵⁵ [i]