MinT - Best Known (240−91, 240, s)-Nets in Base 4

Best Known (240−91, 240, s)-Nets in Base 4

(240−91, 240, 139)-Net over F₄ — Constructive and digital

Digital (149, 240, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 46, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (103, 194, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 97, 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]

(240−91, 240, 152)-Net in Base 4 — Constructive

(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]

(240−91, 240, 411)-Net over F₄ — Digital

Digital (149, 240, 411)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−91, 240, 9224)-Net in Base 4 — Upper bound on s

There is no (149, 240, 9225)-net in base 4, because

1 times m-reduction [i] would yield (149, 239, 9225)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 783863 342342 787942 067373 647821 957993 652466 196288 927896 037004 946793 960729 180342 483966 970320 064469 379172 296397 043390 432853 299226 879088 598878 517248 > 4²³⁹ [i]