MinT - Best Known (248−63, 248, s)-Nets in Base 4

Best Known (248−63, 248, s)-Nets in Base 4

(248−63, 248, 552)-Net over F₄ — Constructive and digital

Digital (185, 248, 552)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 38, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (147, 210, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 70, 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]

(248−63, 248, 648)-Net in Base 4 — Constructive

(185, 248, 648)-net in base 4, using

4 times m-reduction [i] based on (185, 252, 648)-net in base 4, using
- trace code for nets [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]

(248−63, 248, 2073)-Net over F₄ — Digital

Digital (185, 248, 2073)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−63, 248, 259374)-Net in Base 4 — Upper bound on s

There is no (185, 248, 259375)-net in base 4, because

1 times m-reduction [i] would yield (185, 247, 259375)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51147 393029 240119 023077 441355 029928 061118 315895 752400 222390 702598 818215 079170 334189 479600 404629 161486 950287 047507 806238 260315 339780 926246 128132 952376 > 4²⁴⁷ [i]