MinT - Best Known (160−53, 160, s)-Nets in Base 4

Best Known (160−53, 160, s)-Nets in Base 4

(160−53, 160, 195)-Net over F₄ — Constructive and digital

Digital (107, 160, 195)-net over F₄, using

4¹ times duplication [i] based on digital (106, 159, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 53, 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]

(160−53, 160, 240)-Net in Base 4 — Constructive

(107, 160, 240)-net in base 4, using

trace code for nets [i] based on (27, 80, 120)-net in base 16, using
- base change [i] based on digital (11, 64, 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]

(160−53, 160, 465)-Net over F₄ — Digital

Digital (107, 160, 465)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(160−53, 160, 16883)-Net in Base 4 — Upper bound on s

There is no (107, 160, 16884)-net in base 4, because

1 times m-reduction [i] would yield (107, 159, 16884)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 534604 559863 237553 353556 004361 868687 713925 677051 556887 765388 949451 194656 537082 510796 256719 574092 > 4¹⁵⁹ [i]