MinT - Best Known (236−61, 236, s)-Nets in Base 4

Best Known (236−61, 236, s)-Nets in Base 4

(236−61, 236, 541)-Net over F₄ — Constructive and digital

Digital (175, 236, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 32, 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 (143, 204, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 68, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(236−61, 236, 648)-Net in Base 4 — Constructive

(175, 236, 648)-net in base 4, using

1 times m-reduction [i] based on (175, 237, 648)-net in base 4, using
- trace code for nets [i] based on (17, 79, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(236−61, 236, 1834)-Net over F₄ — Digital

Digital (175, 236, 1834)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(236−61, 236, 208810)-Net in Base 4 — Upper bound on s

There is no (175, 236, 208811)-net in base 4, because

1 times m-reduction [i] would yield (175, 235, 208811)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3048 648053 456796 306236 757633 775237 402064 169296 975648 695912 884650 703280 017451 395894 986868 424189 258573 216019 445076 541273 410371 731142 557469 818680 > 4²³⁵ [i]