MinT - Best Known (81−19, 81, s)-Nets in Base 3

Best Known (81−19, 81, s)-Nets in Base 3

(81−19, 81, 400)-Net over F₃ — Constructive and digital

Digital (62, 81, 400)-net over F₃, using

3¹ times duplication [i] based on digital (61, 80, 400)-net over F₃, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(81−19, 81, 616)-Net over F₃ — Digital

Digital (62, 81, 616)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁸¹, 616, F₃, 19) (dual of [616, 535, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁸¹, 758, F₃, 19) (dual of [758, 677, 20]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [i] based on
    1. linear OA(3⁷³, 730, F₃, 19) (dual of [730, 657, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
    2. linear OA(3⁴⁹, 730, F₃, 13) (dual of [730, 681, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730Â | 3¹²âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
    3. linear OA(3⁸, 28, F₃, 5) (dual of [28, 20, 6]-code), using
      - dual code (with bound on d by constructionÂ Y1) [i] based on
        linear OA(3²⁰, 28, F₃, 14) (dual of [28, 8, 15]-code), using
        contraction [i] based on linear OA(3⁴⁸, 56, F₃, 29) (dual of [56, 8, 30]-code), using the BCH-code C(I) with length 56Â | 3⁶âˆ’1, defining interval IÂ =Â {0,1,…,28}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
        nonexistence of linear OA(3¹⁹, 23, F₃, 14) (dual of [23, 4, 15]-code), because
        2 times truncation [i] would yield linear OA(3¹⁷, 21, F₃, 12) (dual of [21, 4, 13]-code), but
        â€œHNâ€ bound on codes from Brouwerâ€™s database [i]

(81−19, 81, 36115)-Net in Base 3 — Upper bound on s

There is no (62, 81, 36116)-net in base 3, because

1 times m-reduction [i] would yield (62, 80, 36116)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 147 816678 675275 584523 228467 679209 317929 > 3⁸⁰ [i]

Best Known (81−19, 81, s)-Nets in Base 3

(81−19, 81, 400)-Net over F3 — Constructive and digital

(81−19, 81, 616)-Net over F3 — Digital

(81−19, 81, 36115)-Net in Base 3 — Upper bound on s

(81−19, 81, 400)-Net over F₃ — Constructive and digital

(81−19, 81, 616)-Net over F₃ — Digital