MinT - Best Known (181−64, 181, s)-Nets in Base 4

Best Known (181−64, 181, s)-Nets in Base 4

(181−64, 181, 152)-Net over F₄ — Constructive and digital

Digital (117, 181, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 41, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]
2. digital (76, 140, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 70, 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]

(181−64, 181, 196)-Net in Base 4 — Constructive

(117, 181, 196)-net in base 4, using

4¹ times duplication [i] based on (116, 180, 196)-net in base 4, using
- t-expansion [i] based on (115, 180, 196)-net in base 4, using
  - trace code for nets [i] based on (25, 90, 98)-net in base 16, using
    - base change [i] based on digital (7, 72, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(181−64, 181, 408)-Net over F₄ — Digital

Digital (117, 181, 408)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(181−64, 181, 10817)-Net in Base 4 — Upper bound on s

There is no (117, 181, 10818)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 417925 139398 129746 119216 836236 165255 328071 711376 954043 109370 097397 489728 529043 347066 847709 063082 433900 454710 > 4¹⁸¹ [i]