MinT - Best Known (209−73, 209, s)-Nets in Base 4

Best Known (209−73, 209, s)-Nets in Base 4

(209−73, 209, 163)-Net over F₄ — Constructive and digital

Digital (136, 209, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 51, 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 (85, 158, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 79, 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]

(209−73, 209, 208)-Net in Base 4 — Constructive

(136, 209, 208)-net in base 4, using

t-expansion [i] based on (135, 209, 208)-net in base 4, using
- 1 times m-reduction [i] based on (135, 210, 208)-net in base 4, using
  - trace code for nets [i] based on (30, 105, 104)-net in base 16, using
    - base change [i] based on digital (9, 84, 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]

(209−73, 209, 484)-Net over F₄ — Digital

Digital (136, 209, 484)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−73, 209, 14298)-Net in Base 4 — Upper bound on s

There is no (136, 209, 14299)-net in base 4, because

1 times m-reduction [i] would yield (136, 208, 14299)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169445 806712 386640 481481 773858 533943 667195 836284 626046 868143 741015 730225 248193 724869 838211 686263 350889 875975 829549 622318 439956 > 4²⁰⁸ [i]