MinT - Best Known (197−31, 197, s)-Nets in Base 3

Best Known (197−31, 197, s)-Nets in Base 3

(197−31, 197, 1480)-Net over F₃ — Constructive and digital

Digital (166, 197, 1480)-net over F₃, using

3 times m-reduction [i] based on digital (166, 200, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 50, 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]

(197−31, 197, 9872)-Net over F₃ — Digital

Digital (166, 197, 9872)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹⁷, 9872, F₃, 2, 31) (dual of [(9872, 2), 19547, 32]-NRT-code), using
- OOA 2-folding [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]

(197−31, 197, 5509597)-Net in Base 3 — Upper bound on s

There is no (166, 197, 5509598)-net in base 3, because

1 times m-reduction [i] would yield (166, 196, 5509598)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3279 186770 159678 938866 771460 871416 490569 967317 155223 972646 517773 507360 483112 581969 537415 378921 > 3¹⁹⁶ [i]