MinT - Best Known (193−69, 193, s)-Nets in Base 4

Best Known (193−69, 193, s)-Nets in Base 4

(193−69, 193, 152)-Net over F₄ — Constructive and digital

Digital (124, 193, 152)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (9, 43, 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 (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]

(193−69, 193, 196)-Net in Base 4 — Constructive

(124, 193, 196)-net in base 4, using

1 times m-reduction [i] based on (124, 194, 196)-net in base 4, using
- trace code for nets [i] based on (27, 97, 98)-net in base 16, using
  - 3 times m-reduction [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]

(193−69, 193, 416)-Net over F₄ — Digital

Digital (124, 193, 416)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(193−69, 193, 11301)-Net in Base 4 — Upper bound on s

There is no (124, 193, 11302)-net in base 4, because

1 times m-reduction [i] would yield (124, 192, 11302)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 39 402484 498450 867905 795098 892307 289945 095679 817308 159478 778297 192028 092635 550382 201685 644379 801451 865302 126987 335730 > 4¹⁹² [i]