MinT - Best Known (160−55, 160, s)-Nets in Base 4

Best Known (160−55, 160, s)-Nets in Base 4

(160−55, 160, 157)-Net over F₄ — Constructive and digital

Digital (105, 160, 157)-net over F₄, using

4¹ times duplication [i] based on digital (104, 159, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 37, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (67, 122, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 61, 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]

(160−55, 160, 208)-Net in Base 4 — Constructive

(105, 160, 208)-net in base 4, using

trace code for nets [i] based on (25, 80, 104)-net in base 16, using
- base change [i] based on digital (9, 64, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(160−55, 160, 406)-Net over F₄ — Digital

Digital (105, 160, 406)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(160−55, 160, 12764)-Net in Base 4 — Upper bound on s

There is no (105, 160, 12765)-net in base 4, because

1 times m-reduction [i] would yield (105, 159, 12765)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 534603 374090 714999 071822 082072 529428 964283 769334 801865 457044 747848 951556 121934 784488 740948 707324 > 4¹⁵⁹ [i]