MinT - Best Known (175−59, 175, s)-Nets in Base 4

Best Known (175−59, 175, s)-Nets in Base 4

(175−59, 175, 163)-Net over F₄ — Constructive and digital

Digital (116, 175, 163)-net over F₄, using

4¹ times duplication [i] based on digital (115, 174, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 44, 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 (71, 130, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 65, 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]

(175−59, 175, 208)-Net in Base 4 — Constructive

(116, 175, 208)-net in base 4, using

3 times m-reduction [i] based on (116, 178, 208)-net in base 4, using
- trace code for nets [i] based on (27, 89, 104)-net in base 16, using
  - 1 times m-reduction [i] based on (27, 90, 104)-net in base 16, using
    - base change [i] based on digital (9, 72, 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]

(175−59, 175, 470)-Net over F₄ — Digital

Digital (116, 175, 470)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(175−59, 175, 15911)-Net in Base 4 — Upper bound on s

There is no (116, 175, 15912)-net in base 4, because

1 times m-reduction [i] would yield (116, 174, 15912)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 573 724978 753021 250696 189609 537315 411026 099031 350123 608752 248669 483432 959471 106409 308739 816950 214941 932748 > 4¹⁷⁴ [i]