MinT - Best Known (133−45, 133, s)-Nets in Base 4

Best Known (133−45, 133, s)-Nets in Base 4

(133−45, 133, 152)-Net over F₄ — Constructive and digital

Digital (88, 133, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 31, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]
2. digital (57, 102, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 51, 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]

(133−45, 133, 196)-Net in Base 4 — Constructive

(88, 133, 196)-net in base 4, using

1 times m-reduction [i] based on (88, 134, 196)-net in base 4, using
- trace code for nets [i] based on (21, 67, 98)-net in base 16, using
  - 3 times m-reduction [i] based on (21, 70, 98)-net in base 16, using
    - base change [i] based on digital (7, 56, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(133−45, 133, 369)-Net over F₄ — Digital

Digital (88, 133, 369)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(133−45, 133, 12344)-Net in Base 4 — Upper bound on s

There is no (88, 133, 12345)-net in base 4, because

1 times m-reduction [i] would yield (88, 132, 12345)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 29 681760 660607 387967 012785 411711 312422 199253 907515 694384 136562 633921 522604 711004 > 4¹³² [i]