MinT - Best Known (99, 150, s)-Nets in Base 4

Best Known (99, 150, s)-Nets in Base 4

(99, 150, 157)-Net over F₄ — Constructive and digital

Digital (99, 150, 157)-net over F₄, using

4¹ times duplication [i] based on digital (98, 149, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 35, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (63, 114, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 57, 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]

(99, 150, 208)-Net in Base 4 — Constructive

(99, 150, 208)-net in base 4, using

trace code for nets [i] based on (24, 75, 104)-net in base 16, using
- base change [i] based on digital (9, 60, 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]

(99, 150, 400)-Net over F₄ — Digital

Digital (99, 150, 400)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 150, 13125)-Net in Base 4 — Upper bound on s

There is no (99, 150, 13126)-net in base 4, because

1 times m-reduction [i] would yield (99, 149, 13126)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 509907 834461 653086 288578 933953 495340 772967 434051 006121 993845 031534 853668 045056 518212 130824 > 4¹⁴⁹ [i]