MinT - Best Known (166−51, 166, s)-Nets in Base 4

Best Known (166−51, 166, s)-Nets in Base 4

(166−51, 166, 312)-Net over F₄ — Constructive and digital

Digital (115, 166, 312)-net over F₄, using

2 times m-reduction [i] based on digital (115, 168, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 56, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(166−51, 166, 644)-Net over F₄ — Digital

Digital (115, 166, 644)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−51, 166, 31902)-Net in Base 4 — Upper bound on s

There is no (115, 166, 31903)-net in base 4, because

1 times m-reduction [i] would yield (115, 165, 31903)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2188 214087 823413 558067 339807 502536 446096 467890 552010 092574 624096 673188 105867 246094 444509 582042 355242 > 4¹⁶⁵ [i]