MinT - Best Known (122, 122+68, s)-Nets in Base 4

Best Known (122, 122+68, s)-Nets in Base 4

(122, 122+68, 151)-Net over F₄ — Constructive and digital

Digital (122, 190, 151)-net over F₄, using

1 times m-reduction [i] based on digital (122, 191, 151)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 41, 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 (81, 150, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 75, 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, 122+68, 196)-Net in Base 4 — Constructive

(122, 190, 196)-net in base 4, using

t-expansion [i] based on (121, 190, 196)-net in base 4, using
- trace code for nets [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]

(122, 122+68, 409)-Net over F₄ — Digital

Digital (122, 190, 409)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(122, 122+68, 10414)-Net in Base 4 — Upper bound on s

There is no (122, 190, 10415)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 464572 662237 550174 849997 056188 093397 326056 393979 757923 853986 460351 639660 334471 298331 232547 800618 611851 320456 608355 > 4¹⁹⁰ [i]