MinT - Best Known (137, 137+53, s)-Nets in Base 4

Best Known (137, 137+53, s)-Nets in Base 4

(137, 137+53, 531)-Net over F₄ — Constructive and digital

Digital (137, 190, 531)-net over F₄, using

5 times m-reduction [i] based on digital (137, 195, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 65, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(137, 137+53, 1089)-Net over F₄ — Digital

Digital (137, 190, 1089)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+53, 83670)-Net in Base 4 — Upper bound on s

There is no (137, 190, 83671)-net in base 4, because

1 times m-reduction [i] would yield (137, 189, 83671)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 615656 416548 306254 425184 003110 890807 940438 292692 254994 251222 357022 357634 715870 417539 155791 740870 965289 718145 989208 > 4¹⁸⁹ [i]