MinT - Best Known (109−37, 109, s)-Nets in Base 4

Best Known (109−37, 109, s)-Nets in Base 4

(109−37, 109, 147)-Net over F₄ — Constructive and digital

Digital (72, 109, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 23, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (49, 86, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 43, 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]

(109−37, 109, 152)-Net in Base 4 — Constructive

(72, 109, 152)-net in base 4, using

t-expansion [i] based on (71, 109, 152)-net in base 4, using
- 1 times m-reduction [i] based on (71, 110, 152)-net in base 4, using
  - trace code for nets [i] based on (16, 55, 76)-net in base 16, using
    - base change [i] based on digital (5, 44, 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]

(109−37, 109, 312)-Net over F₄ — Digital

Digital (72, 109, 312)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−37, 109, 10298)-Net in Base 4 — Upper bound on s

There is no (72, 109, 10299)-net in base 4, because

1 times m-reduction [i] would yield (72, 108, 10299)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 105473 873331 388865 021200 754806 345130 852150 219512 092064 365982 193677 > 4¹⁰⁸ [i]