MinT - Best Known (258−67, 258, s)-Nets in Base 4

Best Known (258−67, 258, s)-Nets in Base 4

(258−67, 258, 545)-Net over F₄ — Constructive and digital

Digital (191, 258, 545)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 36, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, 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]

(258−67, 258, 648)-Net in Base 4 — Constructive

(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]

(258−67, 258, 1945)-Net over F₄ — Digital

Digital (191, 258, 1945)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−67, 258, 214263)-Net in Base 4 — Upper bound on s

There is no (191, 258, 214264)-net in base 4, because

1 times m-reduction [i] would yield (191, 257, 214264)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53634 640145 246380 349712 635220 131276 858348 509982 367034 964298 538211 850848 495801 682562 771573 157833 345779 601735 565553 113770 473583 239945 233338 160265 345494 912605 > 4²⁵⁷ [i]