MinT - Best Known (145, 145+82, s)-Nets in Base 4

Best Known (145, 145+82, s)-Nets in Base 4

(145, 145+82, 157)-Net over F₄ — Constructive and digital

Digital (145, 227, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 51, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (94, 176, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 88, 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]

(145, 145+82, 196)-Net in Base 4 — Constructive

(145, 227, 196)-net in base 4, using

3 times m-reduction [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]

(145, 145+82, 463)-Net over F₄ — Digital

Digital (145, 227, 463)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(145, 145+82, 11558)-Net in Base 4 — Upper bound on s

There is no (145, 227, 11559)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46666 188581 216066 174953 707630 400149 044408 518921 391664 308750 037621 543730 861101 000296 736884 460757 821414 852674 521812 196447 509008 539853 360320 > 4²²⁷ [i]