MinT - Best Known (231−77, 231, s)-Nets in Base 4

Best Known (231−77, 231, s)-Nets in Base 4

(231−77, 231, 195)-Net over F₄ — Constructive and digital

Digital (154, 231, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 77, 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]

(231−77, 231, 240)-Net in Base 4 — Constructive

(154, 231, 240)-net in base 4, using

7 times m-reduction [i] based on (154, 238, 240)-net in base 4, using
- trace code for nets [i] based on (35, 119, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
    - base change [i] based on digital (11, 96, 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]

(231−77, 231, 624)-Net over F₄ — Digital

Digital (154, 231, 624)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−77, 231, 22035)-Net in Base 4 — Upper bound on s

There is no (154, 231, 22036)-net in base 4, because

1 times m-reduction [i] would yield (154, 230, 22036)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 981566 135849 791282 577891 202842 941133 678699 280443 317357 968376 873868 977171 410705 129299 549006 381224 768875 417998 072968 679236 579335 599732 548544 > 4²³⁰ [i]