MinT - Best Known (210−78, 210, s)-Nets in Base 4

Best Known (210−78, 210, s)-Nets in Base 4

(210−78, 210, 144)-Net over F₄ — Constructive and digital

Digital (132, 210, 144)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 42, 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 (90, 168, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 84, 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]

(210−78, 210, 152)-Net in Base 4 — Constructive

(132, 210, 152)-net in base 4, using

t-expansion [i] based on (131, 210, 152)-net in base 4, using
- trace code for nets [i] based on (26, 105, 76)-net in base 16, using
  - base change [i] based on digital (5, 84, 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]

(210−78, 210, 392)-Net over F₄ — Digital

Digital (132, 210, 392)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−78, 210, 8925)-Net in Base 4 — Upper bound on s

There is no (132, 210, 8926)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 718268 807900 117939 697286 793398 766663 348565 958850 018871 981632 547362 531102 405333 081968 422705 747376 982653 321818 445261 953935 491160 > 4²¹⁰ [i]