MinT - Best Known (123, 123+67, s)-Nets in Base 4

Best Known (123, 123+67, s)-Nets in Base 4

(123, 123+67, 157)-Net over F₄ — Constructive and digital

Digital (123, 190, 157)-net over F₄, using

4¹ times duplication [i] based on digital (122, 189, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 43, 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 (79, 146, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 73, 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]

(123, 123+67, 208)-Net in Base 4 — Constructive

(123, 190, 208)-net in base 4, using

trace code for nets [i] based on (28, 95, 104)-net in base 16, using
- base change [i] based on digital (9, 76, 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]

(123, 123+67, 430)-Net over F₄ — Digital

Digital (123, 190, 430)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+67, 12286)-Net in Base 4 — Upper bound on s

There is no (123, 190, 12287)-net in base 4, because

1 times m-reduction [i] would yield (123, 189, 12287)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 615830 523704 277859 311625 800337 121507 369541 324885 592039 029223 848212 065341 005084 076588 403174 927438 038134 922232 912030 > 4¹⁸⁹ [i]