MinT - Best Known (127, 127+76, s)-Nets in Base 4

Best Known (127, 127+76, s)-Nets in Base 4

(127, 127+76, 139)-Net over F₄ — Constructive and digital

Digital (127, 203, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 39, 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 (88, 164, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 82, 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]

(127, 127+76, 370)-Net over F₄ — Digital

Digital (127, 203, 370)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+76, 8209)-Net in Base 4 — Upper bound on s

There is no (127, 203, 8210)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 165 793445 518021 236594 771849 450820 815135 196551 645863 331201 021946 041206 793772 532450 314362 584426 029105 930673 018859 757315 287040 > 4²⁰³ [i]