MinT - Best Known (138−51, 138, s)-Nets in Base 4

Best Known (138−51, 138, s)-Nets in Base 4

(138−51, 138, 130)-Net over F₄ — Constructive and digital

Digital (87, 138, 130)-net over F₄, using

24 times m-reduction [i] based on digital (87, 162, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 81, 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]

(138−51, 138, 278)-Net over F₄ — Digital

Digital (87, 138, 278)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−51, 138, 6737)-Net in Base 4 — Upper bound on s

There is no (87, 138, 6738)-net in base 4, because

1 times m-reduction [i] would yield (87, 137, 6738)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 30432 531336 405300 369237 754913 073661 334268 838882 321597 852634 811848 843855 218005 038296 > 4¹³⁷ [i]