MinT - Best Known (129, 129+72, s)-Nets in Base 4

Best Known (129, 129+72, s)-Nets in Base 4

(129, 129+72, 152)-Net over F₄ — Constructive and digital

Digital (129, 201, 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 (84, 156, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 78, 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]

(129, 129+72, 196)-Net in Base 4 — Constructive

(129, 201, 196)-net in base 4, using

4¹ times duplication [i] based on (128, 200, 196)-net in base 4, using
- t-expansion [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]

(129, 129+72, 428)-Net over F₄ — Digital

Digital (129, 201, 428)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+72, 10913)-Net in Base 4 — Upper bound on s

There is no (129, 201, 10914)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 361227 430133 902235 659613 902400 475719 793682 224107 698728 941100 877689 209499 437830 929409 526730 679804 949297 039074 630365 952608 > 4²⁰¹ [i]