MinT - Best Known (201−71, 201, s)-Nets in Base 4

Best Known (201−71, 201, s)-Nets in Base 4

(201−71, 201, 158)-Net over F₄ — Constructive and digital

Digital (130, 201, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 47, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, 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]

(201−71, 201, 208)-Net in Base 4 — Constructive

(130, 201, 208)-net in base 4, using

4¹ times duplication [i] based on (129, 200, 208)-net in base 4, using
- trace code for nets [i] based on (29, 100, 104)-net in base 16, using
  - base change [i] based on digital (9, 80, 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]

(201−71, 201, 449)-Net over F₄ — Digital

Digital (130, 201, 449)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(201−71, 201, 12750)-Net in Base 4 — Upper bound on s

There is no (130, 201, 12751)-net in base 4, because

1 times m-reduction [i] would yield (130, 200, 12751)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 589198 585856 230840 777448 408730 514588 560220 105853 715282 124823 737203 124440 799479 478800 116065 269623 412538 102268 158830 076304 > 4²⁰⁰ [i]