MinT - Best Known (138−90, 138, s)-Nets in Base 3

Best Known (138−90, 138, s)-Nets in Base 3

(138−90, 138, 48)-Net over F₃ — Constructive and digital

Digital (48, 138, 48)-net over F₃, using

t-expansion [i] based on digital (45, 138, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(138−90, 138, 56)-Net over F₃ — Digital

Digital (48, 138, 56)-net over F₃, using

t-expansion [i] based on digital (40, 138, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(138−90, 138, 160)-Net over F₃ — Upper bound on s (digital)

There is no digital (48, 138, 161)-net over F₃, because

1 times m-reduction [i] would yield digital (48, 137, 161)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹³⁷, 161, F₃, 89) (dual of [161, 24, 90]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(3¹³⁶, 149, S₃, 89), but
      - the linear programming bound shows that MÂ â‰¥ 6 385896 362423 839893 456186 755923 270794 744764 572661 578917 513381 685836 804369 / 75 686875 > 3¹³⁶ [i]
    2. OA(3²⁴, 161, S₃, 12), but
      - discarding factors would yield OA(3²⁴, 124, S₃, 12), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 293147 842993 > 3²⁴ [i]

(138−90, 138, 214)-Net in Base 3 — Upper bound on s

There is no (48, 138, 215)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 761406 402801 083451 328224 989576 494861 855609 325159 299021 649778 567191 > 3¹³⁸ [i]