MinT - Best Known (109, 109+64, s)-Nets in Base 4

Best Known (109, 109+64, s)-Nets in Base 4

(109, 109+64, 139)-Net over F₄ — Constructive and digital

Digital (109, 173, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 33, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [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]

(109, 109+64, 337)-Net over F₄ — Digital

Digital (109, 173, 337)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+64, 7641)-Net in Base 4 — Upper bound on s

There is no (109, 173, 7642)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 143 805579 741981 103677 806230 583132 298348 032894 322448 523732 803626 316876 776182 899908 061973 009451 686220 672550 > 4¹⁷³ [i]