MinT - Best Known (176−62, 176, s)-Nets in Base 4

Best Known (176−62, 176, s)-Nets in Base 4

(176−62, 176, 152)-Net over F₄ — Constructive and digital

Digital (114, 176, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 40, 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 (74, 136, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 68, 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]

(176−62, 176, 196)-Net in Base 4 — Constructive

(114, 176, 196)-net in base 4, using

2 times m-reduction [i] based on (114, 178, 196)-net in base 4, using
- trace code for nets [i] based on (25, 89, 98)-net in base 16, using
  - 1 times m-reduction [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]

(176−62, 176, 404)-Net over F₄ — Digital

Digital (114, 176, 404)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(176−62, 176, 10815)-Net in Base 4 — Upper bound on s

There is no (114, 176, 10816)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9189 090646 866156 258572 199062 358441 324328 209550 727093 189089 984522 186054 817573 783726 627080 261043 663763 769405 > 4¹⁷⁶ [i]