MinT - Best Known (127−37, 127, s)-Nets in Base 4

Best Known (127−37, 127, s)-Nets in Base 4

(127−37, 127, 384)-Net over F₄ — Constructive and digital

Digital (90, 127, 384)-net over F₄, using

4¹ times duplication [i] based on digital (89, 126, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 42, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(127−37, 127, 651)-Net over F₄ — Digital

Digital (90, 127, 651)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−37, 127, 41235)-Net in Base 4 — Upper bound on s

There is no (90, 127, 41236)-net in base 4, because

1 times m-reduction [i] would yield (90, 126, 41236)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7237 142051 039802 533953 181460 846849 895888 616326 202368 653293 881627 736523 208915 > 4¹²⁶ [i]