MinT - Best Known (249−89, 249, s)-Nets in Base 4

Best Known (249−89, 249, s)-Nets in Base 4

(249−89, 249, 163)-Net over F₄ — Constructive and digital

Digital (160, 249, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 59, 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 (101, 190, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 95, 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]

(249−89, 249, 208)-Net in Base 4 — Constructive

(160, 249, 208)-net in base 4, using

t-expansion [i] based on (159, 249, 208)-net in base 4, using
- 1 times m-reduction [i] based on (159, 250, 208)-net in base 4, using
  - trace code for nets [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]

(249−89, 249, 521)-Net over F₄ — Digital

Digital (160, 249, 521)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−89, 249, 14194)-Net in Base 4 — Upper bound on s

There is no (160, 249, 14195)-net in base 4, because

1 times m-reduction [i] would yield (160, 248, 14195)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204772 136696 713319 680108 147293 808875 747133 961071 336159 289278 572488 466940 512486 408886 774318 406475 070251 894841 888283 479402 192812 114363 670429 250076 179620 > 4²⁴⁸ [i]