MinT - Best Known (127, 127+63, s)-Nets in Base 4

Best Known (127, 127+63, s)-Nets in Base 4

(127, 127+63, 195)-Net over F₄ — Constructive and digital

Digital (127, 190, 195)-net over F₄, using

4¹ times duplication [i] based on digital (126, 189, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 63, 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]

(127, 127+63, 240)-Net in Base 4 — Constructive

(127, 190, 240)-net in base 4, using

t-expansion [i] based on (125, 190, 240)-net in base 4, using
- trace code for nets [i] based on (30, 95, 120)-net in base 16, using
  - base change [i] based on digital (11, 76, 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]

(127, 127+63, 536)-Net over F₄ — Digital

Digital (127, 190, 536)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+63, 19362)-Net in Base 4 — Upper bound on s

There is no (127, 190, 19363)-net in base 4, because

1 times m-reduction [i] would yield (127, 189, 19363)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 615861 574993 876243 144878 154843 176569 162025 069781 501949 365017 364207 676927 699742 426754 004874 349445 405694 455859 518080 > 4¹⁸⁹ [i]