MinT - Best Known (154−51, 154, s)-Nets in Base 4

Best Known (154−51, 154, s)-Nets in Base 4

(154−51, 154, 195)-Net over F₄ — Constructive and digital

Digital (103, 154, 195)-net over F₄, using

4¹ times duplication [i] based on digital (102, 153, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 51, 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]

(154−51, 154, 208)-Net in Base 4 — Constructive

(103, 154, 208)-net in base 4, using

2 times m-reduction [i] based on (103, 156, 208)-net in base 4, using
- trace code for nets [i] based on (25, 78, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (25, 80, 104)-net in base 16, using
    - base change [i] based on digital (9, 64, 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]

(154−51, 154, 451)-Net over F₄ — Digital

Digital (103, 154, 451)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(154−51, 154, 16389)-Net in Base 4 — Upper bound on s

There is no (103, 154, 16390)-net in base 4, because

1 times m-reduction [i] would yield (103, 153, 16390)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 130 411033 646309 638685 059150 848556 768112 764896 077263 716518 391868 527783 161415 944115 371116 693504 > 4¹⁵³ [i]