MinT - Best Known (151−54, 151, s)-Nets in Base 4

Best Known (151−54, 151, s)-Nets in Base 4

(151−54, 151, 145)-Net over F₄ — Constructive and digital

Digital (97, 151, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 31, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (66, 120, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 60, 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]

(151−54, 151, 152)-Net in Base 4 — Constructive

(97, 151, 152)-net in base 4, using

4¹ times duplication [i] based on (96, 150, 152)-net in base 4, using
- t-expansion [i] based on (95, 150, 152)-net in base 4, using
  - trace code for nets [i] based on (20, 75, 76)-net in base 16, using
    - base change [i] based on digital (5, 60, 76)-net over F₃₂, using
      - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(151−54, 151, 336)-Net over F₄ — Digital

Digital (97, 151, 336)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(151−54, 151, 8457)-Net in Base 4 — Upper bound on s

There is no (97, 151, 8458)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 164860 230549 458970 730757 862623 526442 958729 188455 141950 692095 949159 427158 965787 113826 657664 > 4¹⁵¹ [i]