MinT - Best Known (122−43, 122, s)-Nets in Base 4

Best Known (122−43, 122, s)-Nets in Base 4

(122−43, 122, 144)-Net over F₄ — Constructive and digital

Digital (79, 122, 144)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (55, 98, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 49, 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]

(122−43, 122, 152)-Net in Base 4 — Constructive

(79, 122, 152)-net in base 4, using

4² times duplication [i] based on (77, 120, 152)-net in base 4, using
- trace code for nets [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]

(122−43, 122, 296)-Net over F₄ — Digital

Digital (79, 122, 296)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(122−43, 122, 8501)-Net in Base 4 — Upper bound on s

There is no (79, 122, 8502)-net in base 4, because

1 times m-reduction [i] would yield (79, 121, 8502)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 068860 352502 709893 236088 048359 711084 696508 426096 802453 800978 319517 650017 > 4¹²¹ [i]