MinT - Best Known (144−50, 144, s)-Nets in Base 4

Best Known (144−50, 144, s)-Nets in Base 4

(144−50, 144, 151)-Net over F₄ — Constructive and digital

Digital (94, 144, 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 (62, 112, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 56, 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]

(144−50, 144, 196)-Net in Base 4 — Constructive

(94, 144, 196)-net in base 4, using

trace code for nets [i] based on (22, 72, 98)-net in base 16, using
- 3 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]

(144−50, 144, 358)-Net over F₄ — Digital

Digital (94, 144, 358)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−50, 144, 9942)-Net in Base 4 — Upper bound on s

There is no (94, 144, 9943)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 498 293002 925410 253031 148277 363642 580970 088638 423819 224520 089863 741846 203783 536824 841520 > 4¹⁴⁴ [i]