MinT - Best Known (178−66, 178, s)-Nets in Base 4

Best Known (178−66, 178, s)-Nets in Base 4

(178−66, 178, 139)-Net over F₄ — Constructive and digital

Digital (112, 178, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 34, 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 (78, 144, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 72, 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]

(178−66, 178, 152)-Net in Base 4 — Constructive

(112, 178, 152)-net in base 4, using

trace code for nets [i] based on (23, 89, 76)-net in base 16, using
- 1 times m-reduction [i] based on (23, 90, 76)-net in base 16, using
  - base change [i] based on digital (5, 72, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(178−66, 178, 343)-Net over F₄ — Digital

Digital (112, 178, 343)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−66, 178, 7730)-Net in Base 4 — Upper bound on s

There is no (112, 178, 7731)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 147228 453975 902760 677483 705545 123726 060771 837277 409651 873353 638099 379846 645195 040314 406325 031449 475922 718438 > 4¹⁷⁸ [i]