MinT - Best Known (231−58, 231, s)-Nets in Base 4

Best Known (231−58, 231, s)-Nets in Base 4

(231−58, 231, 552)-Net over F₄ — Constructive and digital

Digital (173, 231, 552)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 36, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (137, 195, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 65, 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]

(231−58, 231, 648)-Net in Base 4 — Constructive

(173, 231, 648)-net in base 4, using

t-expansion [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]

(231−58, 231, 2055)-Net over F₄ — Digital

Digital (173, 231, 2055)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−58, 231, 243039)-Net in Base 4 — Upper bound on s

There is no (173, 231, 243040)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 908548 422012 062003 882495 974270 981967 908667 348002 678742 986191 357829 751498 774737 499070 534161 444956 300477 107501 909439 697759 135322 445148 525147 > 4²³¹ [i]