MinT - Best Known (144, 227, s)-Nets in Base 4

Best Known (144, 227, s)-Nets in Base 4

(144, 227, 151)-Net over F₄ — Constructive and digital

Digital (144, 227, 151)-net over F₄, using

4¹ times duplication [i] based on digital (143, 226, 151)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 48, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (95, 178, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 89, 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]

(144, 227, 196)-Net in Base 4 — Constructive

(144, 227, 196)-net in base 4, using

1 times m-reduction [i] based on (144, 228, 196)-net in base 4, using
- trace code for nets [i] based on (30, 114, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (30, 115, 98)-net in base 16, using
    - base change [i] based on digital (7, 92, 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]

(144, 227, 443)-Net over F₄ — Digital

Digital (144, 227, 443)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(144, 227, 11172)-Net in Base 4 — Upper bound on s

There is no (144, 227, 11173)-net in base 4, because

1 times m-reduction [i] would yield (144, 226, 11173)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11641 679855 747452 067632 802038 041712 409651 916550 581725 302550 614488 101807 277521 955723 474618 466638 261555 506186 807514 573392 250643 134955 361920 > 4²²⁶ [i]