MinT - Best Known (259−65, 259, s)-Nets in Base 4

Best Known (259−65, 259, s)-Nets in Base 4

(259−65, 259, 558)-Net over F₄ — Constructive and digital

Digital (194, 259, 558)-net over F₄, using

4¹ times duplication [i] based on digital (193, 258, 558)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 42, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (151, 216, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 72, 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]

(259−65, 259, 648)-Net in Base 4 — Constructive

(194, 259, 648)-net in base 4, using

4¹ times duplication [i] based on (193, 258, 648)-net in base 4, using
- t-expansion [i] based on (190, 258, 648)-net in base 4, using
  - trace code for nets [i] based on (18, 86, 216)-net in base 64, using
    - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
      - base change [i] based on digital (5, 78, 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]

(259−65, 259, 2279)-Net over F₄ — Digital

Digital (194, 259, 2279)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−65, 259, 304670)-Net in Base 4 — Upper bound on s

There is no (194, 259, 304671)-net in base 4, because

1 times m-reduction [i] would yield (194, 258, 304671)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 214534 760712 343653 880191 690172 464714 077463 979756 881450 007972 563698 780443 207533 742186 088380 919861 002240 798094 424674 245797 469258 618199 068507 416553 102198 826032 > 4²⁵⁸ [i]