MinT - Best Known (123, 123+61, s)-Nets in Base 4

Best Known (123, 123+61, s)-Nets in Base 4

(123, 123+61, 195)-Net over F₄ — Constructive and digital

Digital (123, 184, 195)-net over F₄, using

4¹ times duplication [i] based on digital (122, 183, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 61, 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]

(123, 123+61, 240)-Net in Base 4 — Constructive

(123, 184, 240)-net in base 4, using

2 times m-reduction [i] based on (123, 186, 240)-net in base 4, using
- trace code for nets [i] based on (30, 93, 120)-net in base 16, using
  - 2 times m-reduction [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]

(123, 123+61, 522)-Net over F₄ — Digital

Digital (123, 184, 522)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+61, 18865)-Net in Base 4 — Upper bound on s

There is no (123, 184, 18866)-net in base 4, because

1 times m-reduction [i] would yield (123, 183, 18866)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 150 408624 578506 275794 285693 109076 627615 215567 533556 699498 839400 002747 129999 481260 834028 821876 805038 586852 171568 > 4¹⁸³ [i]