MinT - Best Known (260−67, 260, s)-Nets in Base 4

Best Known (260−67, 260, s)-Nets in Base 4

(260−67, 260, 548)-Net over F₄ — Constructive and digital

Digital (193, 260, 548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 38, 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 (155, 222, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 74, 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]

(260−67, 260, 648)-Net in Base 4 — Constructive

(193, 260, 648)-net in base 4, using

4² times duplication [i] based on (191, 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]

(260−67, 260, 2027)-Net over F₄ — Digital

Digital (193, 260, 2027)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(260−67, 260, 233045)-Net in Base 4 — Upper bound on s

There is no (193, 260, 233046)-net in base 4, because

1 times m-reduction [i] would yield (193, 259, 233046)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 858124 952241 169226 196593 258452 055675 358805 124945 928205 359401 945031 491184 255856 005853 771595 888608 079802 076877 863547 436752 142615 102219 437487 451950 718870 673878 > 4²⁵⁹ [i]