MinT - Best Known (146−51, 146, s)-Nets in Base 4

Best Known (146−51, 146, s)-Nets in Base 4

(146−51, 146, 151)-Net over F₄ — Constructive and digital

Digital (95, 146, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 32, 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 (63, 114, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 57, 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]

(146−51, 146, 196)-Net in Base 4 — Constructive

(95, 146, 196)-net in base 4, using

trace code for nets [i] based on (22, 73, 98)-net in base 16, using
- 2 times m-reduction [i] based on (22, 75, 98)-net in base 16, using
  - base change [i] based on digital (7, 60, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(146−51, 146, 355)-Net over F₄ — Digital

Digital (95, 146, 355)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(146−51, 146, 10510)-Net in Base 4 — Upper bound on s

There is no (95, 146, 10511)-net in base 4, because

1 times m-reduction [i] would yield (95, 145, 10511)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1992 755811 714533 275525 626353 275126 611704 816746 261552 332746 697139 116160 506355 526978 801824 > 4¹⁴⁵ [i]