MinT - Best Known (115−39, 115, s)-Nets in Base 4

Best Known (115−39, 115, s)-Nets in Base 4

(115−39, 115, 147)-Net over F₄ — Constructive and digital

Digital (76, 115, 147)-net over F₄, using

4¹ times duplication [i] based on digital (75, 114, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 24, 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 (51, 90, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 45, 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]

(115−39, 115, 152)-Net in Base 4 — Constructive

(76, 115, 152)-net in base 4, using

3 times m-reduction [i] based on (76, 118, 152)-net in base 4, using
- trace code for nets [i] based on (17, 59, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (17, 60, 76)-net in base 16, using
    - base change [i] based on digital (5, 48, 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]

(115−39, 115, 326)-Net over F₄ — Digital

Digital (76, 115, 326)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(115−39, 115, 10810)-Net in Base 4 — Upper bound on s

There is no (76, 115, 10811)-net in base 4, because

1 times m-reduction [i] would yield (76, 114, 10811)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 431 866296 461154 215335 170683 611546 941684 817949 882185 981102 816178 060740 > 4¹¹⁴ [i]