MinT - Best Known (182, 182+62, s)-Nets in Base 4

Best Known (182, 182+62, s)-Nets in Base 4

(182, 182+62, 548)-Net over F₄ — Constructive and digital

Digital (182, 244, 548)-net over F₄, using

4¹ times duplication [i] based on digital (181, 243, 548)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 36, 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 (145, 207, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 69, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(182, 182+62, 648)-Net in Base 4 — Constructive

(182, 244, 648)-net in base 4, using

t-expansion [i] based on (181, 244, 648)-net in base 4, using
- 2 times m-reduction [i] based on (181, 246, 648)-net in base 4, using
  - trace code for nets [i] based on (17, 82, 216)-net in base 64, using
    - 2 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
      - base change [i] based on digital (5, 72, 216)-net over F₁₂₈, using
        net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(182, 182+62, 2041)-Net over F₄ — Digital

Digital (182, 244, 2041)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(182, 182+62, 226807)-Net in Base 4 — Upper bound on s

There is no (182, 244, 226808)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 799 190260 118202 182089 696931 590737 302888 132045 912992 411011 942900 083161 886506 875232 532035 045574 938627 542877 472657 485739 454580 734813 122326 694452 953700 > 4²⁴⁴ [i]