MinT - Best Known (203−69, 203, s)-Nets in Base 4

Best Known (203−69, 203, s)-Nets in Base 4

(203−69, 203, 163)-Net over F₄ — Constructive and digital

Digital (134, 203, 163)-net over F₄, using

t-expansion [i] based on digital (133, 203, 163)-net over F₄, using
- 1 times m-reduction [i] based on digital (133, 204, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 50, 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 (83, 154, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 77, 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]

(203−69, 203, 240)-Net in Base 4 — Constructive

(134, 203, 240)-net in base 4, using

1 times m-reduction [i] based on (134, 204, 240)-net in base 4, using
- trace code for nets [i] based on (32, 102, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (32, 105, 120)-net in base 16, using
    - base change [i] based on digital (11, 84, 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]

(203−69, 203, 519)-Net over F₄ — Digital

Digital (134, 203, 519)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(203−69, 203, 17005)-Net in Base 4 — Upper bound on s

There is no (134, 203, 17006)-net in base 4, because

1 times m-reduction [i] would yield (134, 202, 17006)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 41 358455 705010 491115 949321 563568 564640 697698 813367 376013 036699 131463 323491 691051 304172 185834 427062 504707 679394 444619 548418 > 4²⁰² [i]