MinT - Best Known (139−46, 139, s)-Nets in Base 4

Best Known (139−46, 139, s)-Nets in Base 4

(139−46, 139, 195)-Net over F₄ — Constructive and digital

Digital (93, 139, 195)-net over F₄, using

4¹ times duplication [i] based on digital (92, 138, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 46, 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]

(139−46, 139, 208)-Net in Base 4 — Constructive

(93, 139, 208)-net in base 4, using

1 times m-reduction [i] based on (93, 140, 208)-net in base 4, using
- trace code for nets [i] based on (23, 70, 104)-net in base 16, using
  - base change [i] based on digital (9, 56, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(139−46, 139, 416)-Net over F₄ — Digital

Digital (93, 139, 416)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−46, 139, 13654)-Net in Base 4 — Upper bound on s

There is no (93, 139, 13655)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 486006 401430 183351 519393 551402 949952 213178 735428 131271 590470 574128 130211 275032 111596 > 4¹³⁹ [i]