MinT - Best Known (231−83, 231, s)-Nets in Base 4

Best Known (231−83, 231, s)-Nets in Base 4

(231−83, 231, 160)-Net over F₄ — Constructive and digital

Digital (148, 231, 160)-net over F₄, using

1 times m-reduction [i] based on digital (148, 232, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 75, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 157, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(231−83, 231, 208)-Net in Base 4 — Constructive

(148, 231, 208)-net in base 4, using

4¹ times duplication [i] based on (147, 230, 208)-net in base 4, using
- trace code for nets [i] based on (32, 115, 104)-net in base 16, using
  - base change [i] based on digital (9, 92, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(231−83, 231, 479)-Net over F₄ — Digital

Digital (148, 231, 479)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−83, 231, 12795)-Net in Base 4 — Upper bound on s

There is no (148, 231, 12796)-net in base 4, because

1 times m-reduction [i] would yield (148, 230, 12796)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 980752 229564 233886 243199 256937 394656 286981 123221 829623 204418 902806 894620 800958 395100 857139 747403 663565 864710 996525 072434 001607 517540 277763 > 4²³⁰ [i]