MinT - Best Known (138, 138+83, s)-Nets in Base 4

Best Known (138, 138+83, s)-Nets in Base 4

(138, 138+83, 140)-Net over F₄ — Constructive and digital

Digital (138, 221, 140)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 43, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (95, 178, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 89, 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]

(138, 138+83, 152)-Net in Base 4 — Constructive

(138, 221, 152)-net in base 4, using

4¹ times duplication [i] based on (137, 220, 152)-net in base 4, using
- trace code for nets [i] based on (27, 110, 76)-net in base 16, using
  - base change [i] based on digital (5, 88, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(138, 138+83, 394)-Net over F₄ — Digital

Digital (138, 221, 394)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+83, 9115)-Net in Base 4 — Upper bound on s

There is no (138, 221, 9116)-net in base 4, because

1 times m-reduction [i] would yield (138, 220, 9116)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 851454 647532 208921 553404 469847 817749 908318 952108 612956 148267 838544 463078 033868 851856 433963 802713 965109 506492 150141 262763 429022 477508 > 4²²⁰ [i]