MinT - Best Known (146, 231, s)-Nets in Base 4

Best Known (146, 231, s)-Nets in Base 4

(146, 231, 151)-Net over F₄ — Constructive and digital

Digital (146, 231, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 49, 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 (97, 182, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 91, 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, 231, 196)-Net in Base 4 — Constructive

(146, 231, 196)-net in base 4, using

4¹ times duplication [i] based on (145, 230, 196)-net in base 4, using
- trace code for nets [i] based on (30, 115, 98)-net in base 16, using
  - base change [i] based on digital (7, 92, 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, 231, 440)-Net over F₄ — Digital

Digital (146, 231, 440)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(146, 231, 10871)-Net in Base 4 — Upper bound on s

There is no (146, 231, 10872)-net in base 4, because

1 times m-reduction [i] would yield (146, 230, 10872)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 979419 211114 061321 779225 717403 275747 841454 952919 660449 161730 686729 837832 122675 450753 938538 798790 862999 430964 875298 057084 442822 254110 164210 > 4²³⁰ [i]