MinT - Best Known (243−87, 243, s)-Nets in Base 4

Best Known (243−87, 243, s)-Nets in Base 4

(243−87, 243, 160)-Net over F₄ — Constructive and digital

Digital (156, 243, 160)-net over F₄, using

13 times m-reduction [i] based on digital (156, 256, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 83, 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, 173, 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]

(243−87, 243, 208)-Net in Base 4 — Constructive

(156, 243, 208)-net in base 4, using

1 times m-reduction [i] based on (156, 244, 208)-net in base 4, using
- trace code for nets [i] based on (34, 122, 104)-net in base 16, using
  - 3 times m-reduction [i] based on (34, 125, 104)-net in base 16, using
    - base change [i] based on digital (9, 100, 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]

(243−87, 243, 507)-Net over F₄ — Digital

Digital (156, 243, 507)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−87, 243, 13726)-Net in Base 4 — Upper bound on s

There is no (156, 243, 13727)-net in base 4, because

1 times m-reduction [i] would yield (156, 242, 13727)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49 978049 665353 359217 471734 183091 803580 618912 641743 353643 084364 368284 851437 503592 542908 618944 596880 906992 205201 690343 166965 589508 810691 814804 652672 > 4²⁴² [i]