MinT - Best Known (229−59, 229, s)-Nets in Base 4

Best Known (229−59, 229, s)-Nets in Base 4

(229−59, 229, 541)-Net over F₄ — Constructive and digital

Digital (170, 229, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 31, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (139, 198, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 66, 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]

(229−59, 229, 648)-Net in Base 4 — Constructive

(170, 229, 648)-net in base 4, using

2 times m-reduction [i] based on (170, 231, 648)-net in base 4, using
- trace code for nets [i] based on (16, 77, 216)-net in base 64, using
  - base change [i] based on digital (5, 66, 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]

(229−59, 229, 1812)-Net over F₄ — Digital

Digital (170, 229, 1812)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−59, 229, 210566)-Net in Base 4 — Upper bound on s

There is no (170, 229, 210567)-net in base 4, because

1 times m-reduction [i] would yield (170, 228, 210567)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 186085 258104 271781 739798 245773 134799 039955 808590 206713 751280 605905 893106 231429 635002 167529 040064 757085 721546 613997 006703 522792 988781 733280 > 4²²⁸ [i]