MinT - Best Known (137−57, 137, s)-Nets in Base 4

Best Known (137−57, 137, s)-Nets in Base 4

(137−57, 137, 130)-Net over F₄ — Constructive and digital

Digital (80, 137, 130)-net over F₄, using

11 times m-reduction [i] based on digital (80, 148, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 74, 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]

(137−57, 137, 186)-Net over F₄ — Digital

Digital (80, 137, 186)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−57, 137, 3140)-Net in Base 4 — Upper bound on s

There is no (80, 137, 3141)-net in base 4, because

1 times m-reduction [i] would yield (80, 136, 3141)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7603 126722 018430 801811 701557 854809 638573 124996 863462 433708 912230 533211 848204 669656 > 4¹³⁶ [i]