MinT - Best Known (193−33, 193, s)-Nets in Base 3

Best Known (193−33, 193, s)-Nets in Base 3

(193−33, 193, 896)-Net over F₃ — Constructive and digital

Digital (160, 193, 896)-net over F₃, using

3 times m-reduction [i] based on digital (160, 196, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 49, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(193−33, 193, 5569)-Net over F₃ — Digital

Digital (160, 193, 5569)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹³, 5569, F₃, 33) (dual of [5569, 5376, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹³, 6579, F₃, 33) (dual of [6579, 6386, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([1,16]) [i] based on
    1. linear OA(3¹⁷⁷, 6562, F₃, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(3¹⁷⁶, 6562, F₃, 16) (dual of [6562, 6386, 17]-code), using the narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(3¹⁶, 17, F₃, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,3)), using
      - dual of repetition code with length 17 [i]

(193−33, 193, 1807007)-Net in Base 3 — Upper bound on s

There is no (160, 193, 1807008)-net in base 3, because

1 times m-reduction [i] would yield (160, 192, 1807008)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40 483788 274154 406776 780387 816585 760586 503766 398610 724827 220710 750110 242324 175377 183454 672897 > 3¹⁹² [i]