MinT - Best Known (161−53, 161, s)-Nets in Base 4

Best Known (161−53, 161, s)-Nets in Base 4

(161−53, 161, 195)-Net over F₄ — Constructive and digital

Digital (108, 161, 195)-net over F₄, using

1 times m-reduction [i] based on digital (108, 162, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 54, 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]

(161−53, 161, 240)-Net in Base 4 — Constructive

(108, 161, 240)-net in base 4, using

4¹ times duplication [i] based on (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]

(161−53, 161, 479)-Net over F₄ — Digital

Digital (108, 161, 479)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(161−53, 161, 17809)-Net in Base 4 — Upper bound on s

There is no (108, 161, 17810)-net in base 4, because

1 times m-reduction [i] would yield (108, 160, 17810)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 138952 863813 619844 649688 143478 090176 416608 512191 767482 672818 901160 578875 877415 723098 553704 896960 > 4¹⁶⁰ [i]