MinT - Best Known (233−83, 233, s)-Nets in Base 4

Best Known (233−83, 233, s)-Nets in Base 4

(233−83, 233, 160)-Net over F₄ — Constructive and digital

Digital (150, 233, 160)-net over F₄, using

5 times m-reduction [i] based on digital (150, 238, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 77, 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, 161, 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]

(233−83, 233, 208)-Net in Base 4 — Constructive

(150, 233, 208)-net in base 4, using

1 times m-reduction [i] based on (150, 234, 208)-net in base 4, using
- trace code for nets [i] based on (33, 117, 104)-net in base 16, using
  - 3 times m-reduction [i] based on (33, 120, 104)-net in base 16, using
    - base change [i] based on digital (9, 96, 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]

(233−83, 233, 498)-Net over F₄ — Digital

Digital (150, 233, 498)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(233−83, 233, 13693)-Net in Base 4 — Upper bound on s

There is no (150, 233, 13694)-net in base 4, because

1 times m-reduction [i] would yield (150, 232, 13694)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 749326 951988 385041 538180 532086 511652 918379 805468 543779 385001 169730 022125 512239 847653 783141 099698 107799 011545 052539 141649 431301 299244 019475 > 4²³² [i]