MinT - Best Known (120, 120+65, s)-Nets in Base 4

Best Known (120, 120+65, s)-Nets in Base 4

(120, 120+65, 157)-Net over F₄ — Constructive and digital

Digital (120, 185, 157)-net over F₄, using

4¹ times duplication [i] based on digital (119, 184, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 42, 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 (77, 142, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 71, 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]

(120, 120+65, 196)-Net in Base 4 — Constructive

(120, 185, 196)-net in base 4, using

3 times m-reduction [i] based on (120, 188, 196)-net in base 4, using
- trace code for nets [i] based on (26, 94, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (26, 95, 98)-net in base 16, using
    - base change [i] based on digital (7, 76, 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]

(120, 120+65, 426)-Net over F₄ — Digital

Digital (120, 185, 426)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(120, 120+65, 12322)-Net in Base 4 — Upper bound on s

There is no (120, 185, 12323)-net in base 4, because

1 times m-reduction [i] would yield (120, 184, 12323)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 602 683526 925254 192914 969409 454914 930473 559591 102397 747291 197304 004169 563603 091541 646814 277182 905397 362253 481265 > 4¹⁸⁴ [i]