MinT - Best Known (205−53, 205, s)-Nets in Base 4

Best Known (205−53, 205, s)-Nets in Base 4

(205−53, 205, 531)-Net over F₄ — Constructive and digital

Digital (152, 205, 531)-net over F₄, using

t-expansion [i] based on digital (151, 205, 531)-net over F₄, using
- 11 times m-reduction [i] based on digital (151, 216, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 72, 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]

(205−53, 205, 648)-Net in Base 4 — Constructive

(152, 205, 648)-net in base 4, using

4¹ times duplication [i] based on (151, 204, 648)-net in base 4, using
- trace code for nets [i] based on (15, 68, 216)-net in base 64, using
  - 2 times m-reduction [i] based on (15, 70, 216)-net in base 64, using
    - base change [i] based on digital (5, 60, 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]

(205−53, 205, 1619)-Net over F₄ — Digital

Digital (152, 205, 1619)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(205−53, 205, 186199)-Net in Base 4 — Upper bound on s

There is no (152, 205, 186200)-net in base 4, because

1 times m-reduction [i] would yield (152, 204, 186200)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 661 132735 927487 636739 878218 935398 109085 012998 257798 935280 134881 709921 725289 123668 094472 959317 359748 598723 819352 938489 242040 > 4²⁰⁴ [i]