MinT - Best Known (137−61, 137, s)-Nets in Base 4

Best Known (137−61, 137, s)-Nets in Base 4

(137−61, 137, 130)-Net over F₄ — Constructive and digital

Digital (76, 137, 130)-net over F₄, using

3 times m-reduction [i] based on digital (76, 140, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 70, 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]

(137−61, 137, 150)-Net over F₄ — Digital

Digital (76, 137, 150)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−61, 137, 2128)-Net in Base 4 — Upper bound on s

There is no (76, 137, 2129)-net in base 4, because

1 times m-reduction [i] would yield (76, 136, 2129)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7650 314595 571963 675054 545849 755777 200361 814654 532825 975148 940303 868224 220597 444576 > 4¹³⁶ [i]