MinT - Best Known (154−52, 154, s)-Nets in Base 4

Best Known (154−52, 154, s)-Nets in Base 4

(154−52, 154, 158)-Net over F₄ — Constructive and digital

Digital (102, 154, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 38, 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 (64, 116, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 58, 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]

(154−52, 154, 208)-Net in Base 4 — Constructive

(102, 154, 208)-net in base 4, using

trace code for nets [i] based on (25, 77, 104)-net in base 16, using
- 3 times m-reduction [i] based on (25, 80, 104)-net in base 16, using
  - base change [i] based on digital (9, 64, 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]

(154−52, 154, 419)-Net over F₄ — Digital

Digital (102, 154, 419)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(154−52, 154, 12927)-Net in Base 4 — Upper bound on s

There is no (102, 154, 12928)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 522 166879 700637 967719 978069 315005 526785 774540 424210 339442 274060 343948 244445 678211 239519 632825 > 4¹⁵⁴ [i]