MinT - Best Known (139−90, 139, s)-Nets in Base 3

Best Known (139−90, 139, s)-Nets in Base 3

(139−90, 139, 48)-Net over F₃ — Constructive and digital

Digital (49, 139, 48)-net over F₃, using

t-expansion [i] based on digital (45, 139, 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]

(139−90, 139, 64)-Net over F₃ — Digital

Digital (49, 139, 64)-net over F₃, using

net from sequence [i] based on digital (49, 63)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
  - a curve from the manYPoints database [i]

(139−90, 139, 186)-Net over F₃ — Upper bound on s (digital)

There is no digital (49, 139, 187)-net over F₃, because

1 times m-reduction [i] would yield digital (49, 138, 187)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹³⁸, 187, F₃, 89) (dual of [187, 49, 90]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹³⁷, 161, F₃, 89) (dual of [161, 24, 90]-code), but
      - constructionÂ Y1 [i] would yield
        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]
        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]
    2. OA(3⁴⁹, 187, S₃, 26), but
      - discarding factors would yield OA(3⁴⁹, 184, S₃, 26), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 245469 510905 440390 144353 > 3⁴⁹ [i]

(139−90, 139, 220)-Net in Base 3 — Upper bound on s

There is no (49, 139, 221)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 178049 164740 036348 013203 822153 715472 696701 392087 985473 930966 278059 > 3¹³⁹ [i]