MinT - Best Known (257−65, 257, s)-Nets in Base 4

Best Known (257−65, 257, s)-Nets in Base 4

(257−65, 257, 553)-Net over F₄ — Constructive and digital

Digital (192, 257, 553)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 41, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, 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]

(257−65, 257, 648)-Net in Base 4 — Constructive

(192, 257, 648)-net in base 4, using

t-expansion [i] based on (190, 257, 648)-net in base 4, using
- 1 times m-reduction [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]

(257−65, 257, 2183)-Net over F₄ — Digital

Digital (192, 257, 2183)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−65, 257, 279381)-Net in Base 4 — Upper bound on s

There is no (192, 257, 279382)-net in base 4, because

1 times m-reduction [i] would yield (192, 256, 279382)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13407 858564 169755 481391 355334 272045 430839 432376 274258 745453 459208 831808 167328 624342 376642 740110 198765 780445 329857 581597 588430 971202 549858 795751 033516 006330 > 4²⁵⁶ [i]