MinT - Best Known (170−62, 170, s)-Nets in Base 4

Best Known (170−62, 170, s)-Nets in Base 4

(170−62, 170, 144)-Net over F₄ — Constructive and digital

Digital (108, 170, 144)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 34, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, 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]

(170−62, 170, 152)-Net in Base 4 — Constructive

(108, 170, 152)-net in base 4, using

t-expansion [i] based on (107, 170, 152)-net in base 4, using
- trace code for nets [i] based on (22, 85, 76)-net in base 16, using
  - base change [i] based on digital (5, 68, 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]

(170−62, 170, 348)-Net over F₄ — Digital

Digital (108, 170, 348)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−62, 170, 8264)-Net in Base 4 — Upper bound on s

There is no (108, 170, 8265)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 246056 431831 623456 081481 832237 053862 587925 262168 066776 793977 430836 092194 460726 349887 831585 805386 838720 > 4¹⁷⁰ [i]