MinT - Best Known (180−51, 180, s)-Nets in Base 4

Best Known (180−51, 180, s)-Nets in Base 4

(180−51, 180, 531)-Net over F₄ — Constructive and digital

Digital (129, 180, 531)-net over F₄, using

3 times m-reduction [i] based on digital (129, 183, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 61, 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]

(180−51, 180, 972)-Net over F₄ — Digital

Digital (129, 180, 972)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−51, 180, 69362)-Net in Base 4 — Upper bound on s

There is no (129, 180, 69363)-net in base 4, because

1 times m-reduction [i] would yield (129, 179, 69363)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 587170 654597 206086 573595 058288 185376 802281 752933 098357 110235 907978 720344 926307 867609 411003 007321 731134 300936 > 4¹⁷⁹ [i]