MinT - Best Known (220−77, 220, s)-Nets in Base 4

Best Known (220−77, 220, s)-Nets in Base 4

(220−77, 220, 163)-Net over F₄ — Constructive and digital

Digital (143, 220, 163)-net over F₄, using

4¹ times duplication [i] based on digital (142, 219, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 53, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (89, 166, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 83, 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]

(220−77, 220, 240)-Net in Base 4 — Constructive

(143, 220, 240)-net in base 4, using

trace code for nets [i] based on (33, 110, 120)-net in base 16, using
- base change [i] based on digital (11, 88, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(220−77, 220, 503)-Net over F₄ — Digital

Digital (143, 220, 503)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−77, 220, 14741)-Net in Base 4 — Upper bound on s

There is no (143, 220, 14742)-net in base 4, because

1 times m-reduction [i] would yield (143, 219, 14742)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 711304 852586 738759 920662 454813 749976 075587 308015 898452 252884 936838 719779 538062 055435 680788 508936 959877 090309 431520 910225 127429 119360 > 4²¹⁹ [i]