MinT - Best Known (136−55, 136, s)-Nets in Base 4

Best Known (136−55, 136, s)-Nets in Base 4

(136−55, 136, 130)-Net over F₄ — Constructive and digital

Digital (81, 136, 130)-net over F₄, using

14 times m-reduction [i] based on digital (81, 150, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 75, 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]

(136−55, 136, 203)-Net over F₄ — Digital

Digital (81, 136, 203)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−55, 136, 3706)-Net in Base 4 — Upper bound on s

There is no (81, 136, 3707)-net in base 4, because

1 times m-reduction [i] would yield (81, 135, 3707)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1897 268486 926404 286571 183637 201710 796082 988893 464382 672712 422484 742878 321071 919860 > 4¹³⁵ [i]