MinT - Best Known (154−55, 154, s)-Nets in Base 4

Best Known (154−55, 154, s)-Nets in Base 4

(154−55, 154, 147)-Net over F₄ — Constructive and digital

Digital (99, 154, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 32, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (67, 122, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 61, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(154−55, 154, 152)-Net in Base 4 — Constructive

(99, 154, 152)-net in base 4, using

2 times m-reduction [i] based on (99, 156, 152)-net in base 4, using
- trace code for nets [i] based on (21, 78, 76)-net in base 16, using
  - 2 times m-reduction [i] based on (21, 80, 76)-net in base 16, using
    - base change [i] based on digital (5, 64, 76)-net over F₃₂, using
      - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(154−55, 154, 343)-Net over F₄ — Digital

Digital (99, 154, 343)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(154−55, 154, 9374)-Net in Base 4 — Upper bound on s

There is no (99, 154, 9375)-net in base 4, because

1 times m-reduction [i] would yield (99, 153, 9375)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 130 609720 050767 955476 817308 143507 760683 877826 074876 233328 112187 534050 166610 364439 905579 147376 > 4¹⁵³ [i]