MinT - Best Known (243−61, 243, s)-Nets in Base 4

Best Known (243−61, 243, s)-Nets in Base 4

(243−61, 243, 553)-Net over F₄ — Constructive and digital

Digital (182, 243, 553)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 39, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, 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]

(243−61, 243, 648)-Net in Base 4 — Constructive

(182, 243, 648)-net in base 4, using

t-expansion [i] based on (181, 243, 648)-net in base 4, using
- 3 times m-reduction [i] based on (181, 246, 648)-net in base 4, using
  - trace code for nets [i] based on (17, 82, 216)-net in base 64, using
    - 2 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]

(243−61, 243, 2151)-Net over F₄ — Digital

Digital (182, 243, 2151)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−61, 243, 288567)-Net in Base 4 — Upper bound on s

There is no (182, 243, 288568)-net in base 4, because

1 times m-reduction [i] would yield (182, 242, 288568)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49 949223 562970 106388 599908 427695 538783 773759 534775 824416 513650 236383 950791 307823 190765 105133 584822 080536 986187 741346 999654 894453 695040 364892 763660 > 4²⁴² [i]