MinT - Best Known (259−70, 259, s)-Nets in Base 4

Best Known (259−70, 259, s)-Nets in Base 4

(259−70, 259, 531)-Net over F₄ — Constructive and digital

Digital (189, 259, 531)-net over F₄, using

4¹ times duplication [i] based on digital (188, 258, 531)-net over F₄, using
- t-expansion [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]

(259−70, 259, 576)-Net in Base 4 — Constructive

(189, 259, 576)-net in base 4, using

4¹ times duplication [i] based on (188, 258, 576)-net in base 4, using
- trace code for nets [i] based on (16, 86, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (16, 91, 192)-net in base 64, using
    - base change [i] based on digital (3, 78, 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]

(259−70, 259, 1627)-Net over F₄ — Digital

Digital (189, 259, 1627)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−70, 259, 132218)-Net in Base 4 — Upper bound on s

There is no (189, 259, 132219)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 858206 166444 254503 334453 110668 419199 936979 125595 566913 817012 179581 385622 717568 684580 373680 130536 913755 992151 677296 779226 623438 872171 598799 640470 932707 760420 > 4²⁵⁹ [i]