MinT - Best Known (139, 139+78, s)-Nets in Base 4

Best Known (139, 139+78, s)-Nets in Base 4

(139, 139+78, 157)-Net over F₄ — Constructive and digital

Digital (139, 217, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 49, 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 (90, 168, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 84, 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]

(139, 139+78, 196)-Net in Base 4 — Constructive

(139, 217, 196)-net in base 4, using

3 times m-reduction [i] based on (139, 220, 196)-net in base 4, using
- trace code for nets [i] based on (29, 110, 98)-net in base 16, using
  - base change [i] based on digital (7, 88, 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]

(139, 139+78, 453)-Net over F₄ — Digital

Digital (139, 217, 453)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+78, 11455)-Net in Base 4 — Upper bound on s

There is no (139, 217, 11456)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44424 872174 095905 451252 457700 736824 641196 924047 123487 837872 210281 177765 176900 041039 743138 464817 807949 044970 183644 849958 591030 162215 > 4²¹⁷ [i]