MinT - Best Known (180−77, 180, s)-Nets in Base 4

Best Known (180−77, 180, s)-Nets in Base 4

(180−77, 180, 130)-Net over F₄ — Constructive and digital

Digital (103, 180, 130)-net over F₄, using

14 times m-reduction [i] based on digital (103, 194, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 97, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(180−77, 180, 215)-Net over F₄ — Digital

Digital (103, 180, 215)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−77, 180, 3402)-Net in Base 4 — Upper bound on s

There is no (103, 180, 3403)-net in base 4, because

1 times m-reduction [i] would yield (103, 179, 3403)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 592625 531965 073353 842347 510572 359387 057867 037320 063716 928616 469816 036719 405327 037524 742929 512846 855426 395740 > 4¹⁷⁹ [i]