MinT - Best Known (111−39, 111, s)-Nets in Base 4

Best Known (111−39, 111, s)-Nets in Base 4

(111−39, 111, 140)-Net over F₄ — Constructive and digital

Digital (72, 111, 140)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 21, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (51, 90, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 45, 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]

(111−39, 111, 152)-Net in Base 4 — Constructive

(72, 111, 152)-net in base 4, using

4¹ times duplication [i] based on (71, 110, 152)-net in base 4, using
- trace code for nets [i] based on (16, 55, 76)-net in base 16, using
  - base change [i] based on digital (5, 44, 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]

(111−39, 111, 278)-Net over F₄ — Digital

Digital (72, 111, 278)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−39, 111, 8070)-Net in Base 4 — Upper bound on s

There is no (72, 111, 8071)-net in base 4, because

1 times m-reduction [i] would yield (72, 110, 8071)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 688413 833532 847653 589297 546496 929893 762306 901562 359578 388897 956096 > 4¹¹⁰ [i]