MinT - Best Known (109, 109+59, s)-Nets in Base 4

Best Known (109, 109+59, s)-Nets in Base 4

(109, 109+59, 152)-Net over F₄ — Constructive and digital

Digital (109, 168, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 38, 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 (71, 130, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 65, 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]

(109, 109+59, 196)-Net in Base 4 — Constructive

(109, 168, 196)-net in base 4, using

2 times m-reduction [i] based on (109, 170, 196)-net in base 4, using
- trace code for nets [i] based on (24, 85, 98)-net in base 16, using
  - base change [i] based on digital (7, 68, 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]

(109, 109+59, 392)-Net over F₄ — Digital

Digital (109, 168, 392)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+59, 11379)-Net in Base 4 — Upper bound on s

There is no (109, 168, 11380)-net in base 4, because

1 times m-reduction [i] would yield (109, 167, 11380)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35014 476445 424473 119862 638598 872812 996885 180876 089539 386652 496502 278414 286445 432670 272537 835523 363944 > 4¹⁶⁷ [i]