MinT - Best Known (225−79, 225, s)-Nets in Base 4

Best Known (225−79, 225, s)-Nets in Base 4

(225−79, 225, 163)-Net over F₄ — Constructive and digital

Digital (146, 225, 163)-net over F₄, using

4¹ times duplication [i] based on digital (145, 224, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 54, 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 (91, 170, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 85, 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]

(225−79, 225, 208)-Net in Base 4 — Constructive

(146, 225, 208)-net in base 4, using

3 times m-reduction [i] based on (146, 228, 208)-net in base 4, using
- trace code for nets [i] based on (32, 114, 104)-net in base 16, using
  - 1 times m-reduction [i] based on (32, 115, 104)-net in base 16, using
    - base change [i] based on digital (9, 92, 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]

(225−79, 225, 507)-Net over F₄ — Digital

Digital (146, 225, 507)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(225−79, 225, 14700)-Net in Base 4 — Upper bound on s

There is no (146, 225, 14701)-net in base 4, because

1 times m-reduction [i] would yield (146, 224, 14701)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 726 891546 194160 782784 716773 178560 818880 409824 070389 307351 553924 653256 263872 964850 680948 738187 038422 482935 383841 056895 543091 479614 718728 > 4²²⁴ [i]