MinT - Best Known (130, 130+73, s)-Nets in Base 4

Best Known (130, 130+73, s)-Nets in Base 4

(130, 130+73, 152)-Net over F₄ — Constructive and digital

Digital (130, 203, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 45, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]
2. digital (85, 158, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 79, 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]

(130, 130+73, 196)-Net in Base 4 — Constructive

(130, 203, 196)-net in base 4, using

1 times m-reduction [i] based on (130, 204, 196)-net in base 4, using
- trace code for nets [i] based on (28, 102, 98)-net in base 16, using
  - 3 times m-reduction [i] based on (28, 105, 98)-net in base 16, using
    - base change [i] based on digital (7, 84, 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]

(130, 130+73, 427)-Net over F₄ — Digital

Digital (130, 203, 427)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(130, 130+73, 11342)-Net in Base 4 — Upper bound on s

There is no (130, 203, 11343)-net in base 4, because

1 times m-reduction [i] would yield (130, 202, 11343)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 41 362210 834436 827650 067661 802670 771594 847181 414587 001972 060026 860984 865924 328274 145167 684672 022068 277294 176882 020124 597390 > 4²⁰² [i]