MinT - Best Known (178, 178+66, s)-Nets in Base 4

Best Known (178, 178+66, s)-Nets in Base 4

(178, 178+66, 531)-Net over F₄ — Constructive and digital

Digital (178, 244, 531)-net over F₄, using

t-expansion [i] based on digital (177, 244, 531)-net over F₄, using
- 11 times m-reduction [i] based on digital (177, 255, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 85, 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]

(178, 178+66, 576)-Net in Base 4 — Constructive

(178, 244, 576)-net in base 4, using

4¹ times duplication [i] based on (177, 243, 576)-net in base 4, using
- trace code for nets [i] based on (15, 81, 192)-net in base 64, using
  - 3 times m-reduction [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]

(178, 178+66, 1541)-Net over F₄ — Digital

Digital (178, 244, 1541)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(178, 178+66, 124089)-Net in Base 4 — Upper bound on s

There is no (178, 244, 124090)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 799 369110 995688 722939 795296 717870 944858 965356 019839 790591 122288 067541 003742 981797 223798 157700 189358 945803 628983 455289 314757 576295 577971 127570 671105 > 4²⁴⁴ [i]