MinT - Best Known (242−86, 242, s)-Nets in Base 4

Best Known (242−86, 242, s)-Nets in Base 4

(242−86, 242, 163)-Net over F₄ — Constructive and digital

Digital (156, 242, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 58, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (98, 184, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 92, 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]

(242−86, 242, 208)-Net in Base 4 — Constructive

(156, 242, 208)-net in base 4, using

2 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]

(242−86, 242, 519)-Net over F₄ — Digital

Digital (156, 242, 519)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−86, 242, 13726)-Net in Base 4 — Upper bound on s

There is no (156, 242, 13727)-net in base 4, because

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]