MinT - Best Known (128, 128+73, s)-Nets in Base 4

Best Known (128, 128+73, s)-Nets in Base 4

(128, 128+73, 151)-Net over F₄ — Constructive and digital

Digital (128, 201, 151)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 43, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, 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]

(128, 128+73, 196)-Net in Base 4 — Constructive

(128, 201, 196)-net in base 4, using

4¹ times duplication [i] based on (127, 200, 196)-net in base 4, using
- trace code for nets [i] based on (27, 100, 98)-net in base 16, using
  - base change [i] based on digital (7, 80, 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]

(128, 128+73, 409)-Net over F₄ — Digital

Digital (128, 201, 409)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+73, 10499)-Net in Base 4 — Upper bound on s

There is no (128, 201, 10500)-net in base 4, because

1 times m-reduction [i] would yield (128, 200, 10500)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 585008 400500 663736 160519 566212 609216 560388 659834 636169 442890 262118 574718 365825 916107 112994 368361 710767 164721 518248 422336 > 4²⁰⁰ [i]