MinT - Best Known (221−81, 221, s)-Nets in Base 4

Best Known (221−81, 221, s)-Nets in Base 4

(221−81, 221, 151)-Net over F₄ — Constructive and digital

Digital (140, 221, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 47, 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 (93, 174, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 87, 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]

(221−81, 221, 196)-Net in Base 4 — Constructive

(140, 221, 196)-net in base 4, using

4¹ times duplication [i] based on (139, 220, 196)-net in base 4, using
- trace code for nets [i] based on (29, 110, 98)-net in base 16, using
  - base change [i] based on digital (7, 88, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(221−81, 221, 430)-Net over F₄ — Digital

Digital (140, 221, 430)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−81, 221, 10731)-Net in Base 4 — Upper bound on s

There is no (140, 221, 10732)-net in base 4, because

1 times m-reduction [i] would yield (140, 220, 10732)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 840767 398133 433539 155946 795600 771305 892815 305120 436079 567381 548067 767467 799263 050324 260852 701265 384097 398692 129950 060004 659156 378183 > 4²²⁰ [i]