MinT - Best Known (241−83, 241, s)-Nets in Base 4

Best Known (241−83, 241, s)-Nets in Base 4

(241−83, 241, 164)-Net over F₄ — Constructive and digital

Digital (158, 241, 164)-net over F₄, using

4¹ times duplication [i] based on digital (157, 240, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 62, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (95, 178, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 89, 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]

(241−83, 241, 240)-Net in Base 4 — Constructive

(158, 241, 240)-net in base 4, using

3 times m-reduction [i] based on (158, 244, 240)-net in base 4, using
- trace code for nets [i] based on (36, 122, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (36, 125, 120)-net in base 16, using
    - base change [i] based on digital (11, 100, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(241−83, 241, 577)-Net over F₄ — Digital

Digital (158, 241, 577)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−83, 241, 17956)-Net in Base 4 — Upper bound on s

There is no (158, 241, 17957)-net in base 4, because

1 times m-reduction [i] would yield (158, 240, 17957)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 122167 350611 067066 154398 720799 509166 277764 618955 842241 830432 429532 160945 716148 570726 614363 062255 564779 945631 733902 332481 229911 434669 710071 403008 > 4²⁴⁰ [i]