MinT - Best Known (169−56, 169, s)-Nets in Base 4

Best Known (169−56, 169, s)-Nets in Base 4

(169−56, 169, 195)-Net over F₄ — Constructive and digital

Digital (113, 169, 195)-net over F₄, using

4¹ times duplication [i] based on digital (112, 168, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 56, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(169−56, 169, 240)-Net in Base 4 — Constructive

(113, 169, 240)-net in base 4, using

1 times m-reduction [i] based on (113, 170, 240)-net in base 4, using
- trace code for nets [i] based on (28, 85, 120)-net in base 16, using
  - base change [i] based on digital (11, 68, 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]

(169−56, 169, 486)-Net over F₄ — Digital

Digital (113, 169, 486)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−56, 169, 16185)-Net in Base 4 — Upper bound on s

There is no (113, 169, 16186)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 560136 943849 071283 609104 642776 233188 462208 274142 838342 255733 655216 507384 424069 888410 850963 198506 527320 > 4¹⁶⁹ [i]