MinT - Best Known (174−59, 174, s)-Nets in Base 4

Best Known (174−59, 174, s)-Nets in Base 4

(174−59, 174, 163)-Net over F₄ — Constructive and digital

Digital (115, 174, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 44, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (71, 130, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 65, 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]

(174−59, 174, 208)-Net in Base 4 — Constructive

(115, 174, 208)-net in base 4, using

2 times m-reduction [i] based on (115, 176, 208)-net in base 4, using
- trace code for nets [i] based on (27, 88, 104)-net in base 16, using
  - 2 times m-reduction [i] based on (27, 90, 104)-net in base 16, using
    - base change [i] based on digital (9, 72, 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]

(174−59, 174, 458)-Net over F₄ — Digital

Digital (115, 174, 458)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(174−59, 174, 15167)-Net in Base 4 — Upper bound on s

There is no (115, 174, 15168)-net in base 4, because

1 times m-reduction [i] would yield (115, 173, 15168)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 143 391389 994616 814289 421080 629837 428879 311239 536094 834296 639787 571948 033854 651081 172126 454684 840180 928077 > 4¹⁷³ [i]