MinT - Best Known (245−63, 245, s)-Nets in Base 4

Best Known (245−63, 245, s)-Nets in Base 4

(245−63, 245, 546)-Net over F₄ — Constructive and digital

Digital (182, 245, 546)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 35, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, 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]

(245−63, 245, 648)-Net in Base 4 — Constructive

(182, 245, 648)-net in base 4, using

t-expansion [i] based on (181, 245, 648)-net in base 4, using
- 1 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]

(245−63, 245, 1941)-Net over F₄ — Digital

Digital (182, 245, 1941)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−63, 245, 226807)-Net in Base 4 — Upper bound on s

There is no (182, 245, 226808)-net in base 4, because

1 times m-reduction [i] would yield (182, 244, 226808)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 799 190260 118202 182089 696931 590737 302888 132045 912992 411011 942900 083161 886506 875232 532035 045574 938627 542877 472657 485739 454580 734813 122326 694452 953700 > 4²⁴⁴ [i]