MinT - Best Known (214−33, 214, s)-Nets in Base 3

Best Known (214−33, 214, s)-Nets in Base 3

(214−33, 214, 1488)-Net over F₃ — Constructive and digital

Digital (181, 214, 1488)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (163, 196, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 49, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(214−33, 214, 11754)-Net over F₃ — Digital

Digital (181, 214, 11754)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹⁴, 11754, F₃, 33) (dual of [11754, 11540, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹⁴, 19743, F₃, 33) (dual of [19743, 19529, 34]-code), using
  - 1 times truncation [i] based on linear OA(3²¹⁵, 19744, F₃, 34) (dual of [19744, 19529, 35]-code), using
    - constructionÂ X applied to C_e(33) âŠ‚ C_e(25) [i] based on
      1. linear OA(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      2. linear OA(3¹⁵⁴, 19683, F₃, 26) (dual of [19683, 19529, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
      3. linear OA(3¹⁶, 61, F₃, 7) (dual of [61, 45, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(214−33, 214, 7641604)-Net in Base 3 — Upper bound on s

There is no (181, 214, 7641605)-net in base 3, because

1 times m-reduction [i] would yield (181, 213, 7641605)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 423475 013408 650960 950923 331216 934391 662023 302723 128938 219684 977474 440428 229021 153555 608110 992152 798657 > 3²¹³ [i]