MinT - Best Known (229−83, 229, s)-Nets in Base 4

Best Known (229−83, 229, s)-Nets in Base 4

(229−83, 229, 157)-Net over F₄ — Constructive and digital

Digital (146, 229, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 51, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (95, 178, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 89, 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]

(229−83, 229, 196)-Net in Base 4 — Constructive

(146, 229, 196)-net in base 4, using

t-expansion [i] based on (145, 229, 196)-net in base 4, using
- 1 times m-reduction [i] based on (145, 230, 196)-net in base 4, using
  - trace code for nets [i] based on (30, 115, 98)-net in base 16, using
    - base change [i] based on digital (7, 92, 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]

(229−83, 229, 461)-Net over F₄ — Digital

Digital (146, 229, 461)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−83, 229, 11956)-Net in Base 4 — Upper bound on s

There is no (146, 229, 11957)-net in base 4, because

1 times m-reduction [i] would yield (146, 228, 11957)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 186236 725242 807523 975876 791441 084988 113513 738269 844828 036438 927816 455293 463218 855881 385064 371966 379891 958404 470496 290928 175636 932451 600688 > 4²²⁸ [i]