MinT - Best Known (155, 155+77, s)-Nets in Base 4

Best Known (155, 155+77, s)-Nets in Base 4

(155, 155+77, 195)-Net over F₄ — Constructive and digital

Digital (155, 232, 195)-net over F₄, using

4¹ times duplication [i] based on 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]

(155, 155+77, 240)-Net in Base 4 — Constructive

(155, 232, 240)-net in base 4, using

8 times m-reduction [i] based on (155, 240, 240)-net in base 4, using
- trace code for nets [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]

(155, 155+77, 636)-Net over F₄ — Digital

Digital (155, 232, 636)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(155, 155+77, 22855)-Net in Base 4 — Upper bound on s

There is no (155, 232, 22856)-net in base 4, because

1 times m-reduction [i] would yield (155, 231, 22856)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 928139 344963 965269 018782 683560 369340 583259 426385 539241 646737 238811 830889 685064 361171 423610 231336 274538 592595 759303 720326 310024 148826 943490 > 4²³¹ [i]