MinT - Best Known (133, 133+71, s)-Nets in Base 4

Best Known (133, 133+71, s)-Nets in Base 4

(133, 133+71, 163)-Net over F₄ — Constructive and digital

Digital (133, 204, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 50, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (83, 154, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 77, 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, 133+71, 208)-Net in Base 4 — Constructive

(133, 204, 208)-net in base 4, using

2 times m-reduction [i] based on (133, 206, 208)-net in base 4, using
- trace code for nets [i] based on (30, 103, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (30, 105, 104)-net in base 16, using
    - base change [i] based on digital (9, 84, 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]

(133, 133+71, 479)-Net over F₄ — Digital

Digital (133, 204, 479)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(133, 133+71, 14362)-Net in Base 4 — Upper bound on s

There is no (133, 204, 14363)-net in base 4, because

1 times m-reduction [i] would yield (133, 203, 14363)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 165 529858 782394 455161 573638 692940 406354 533509 715094 495350 397420 119206 593588 911840 333507 642422 081281 231604 770352 812599 061248 > 4²⁰³ [i]