MinT - Best Known (169, 169+31, s)-Nets in Base 3

Best Known (169, 169+31, s)-Nets in Base 3

(169, 169+31, 1480)-Net over F₃ — Constructive and digital

Digital (169, 200, 1480)-net over F₃, using

4 times m-reduction [i] based on digital (169, 204, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 51, 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]

(169, 169+31, 10940)-Net over F₃ — Digital

Digital (169, 200, 10940)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁰, 10940, F₃, 31) (dual of [10940, 10740, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁰, 19747, F₃, 31) (dual of [19747, 19547, 32]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(3¹⁹⁷, 19744, F₃, 31) (dual of [19744, 19547, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(22) [i] based on
      1. linear OA(3¹⁸¹, 19683, F₃, 31) (dual of [19683, 19502, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(3¹³⁶, 19683, F₃, 23) (dual of [19683, 19547, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [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]

(169, 169+31, 6863480)-Net in Base 3 — Upper bound on s

There is no (169, 200, 6863481)-net in base 3, because

1 times m-reduction [i] would yield (169, 199, 6863481)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 88538 186653 147404 948342 359792 513362 706090 631762 507776 879817 435443 877945 075700 832618 204419 964491 > 3¹⁹⁹ [i]