MinT - Best Known (238−89, 238, s)-Nets in Base 4

Best Known (238−89, 238, s)-Nets in Base 4

(238−89, 238, 145)-Net over F₄ — Constructive and digital

Digital (149, 238, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 48, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (101, 190, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 95, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(238−89, 238, 152)-Net in Base 4 — Constructive

(149, 238, 152)-net in base 4, using

2 times m-reduction [i] based on (149, 240, 152)-net in base 4, using
- trace code for nets [i] based on (29, 120, 76)-net in base 16, using
  - base change [i] based on digital (5, 96, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(238−89, 238, 427)-Net over F₄ — Digital

Digital (149, 238, 427)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−89, 238, 10026)-Net in Base 4 — Upper bound on s

There is no (149, 238, 10027)-net in base 4, because

1 times m-reduction [i] would yield (149, 237, 10027)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48861 497895 191198 692819 800167 419610 830849 122744 684936 943016 724047 318371 304951 960858 042058 833294 018268 330616 071940 071417 017257 060425 665753 322592 > 4²³⁷ [i]