MinT - Best Known (149−31, 149, s)-Nets in Base 3

Best Known (149−31, 149, s)-Nets in Base 3

(149−31, 149, 640)-Net over F₃ — Constructive and digital

Digital (118, 149, 640)-net over F₃, using

3¹ times duplication [i] based on digital (117, 148, 640)-net over F₃, using
- t-expansion [i] based on digital (116, 148, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 37, 160)-net over F₈₁, using
    - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
        a function field by SÃ©mirat [i]

(149−31, 149, 1562)-Net over F₃ — Digital

Digital (118, 149, 1562)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴⁹, 1562, F₃, 31) (dual of [1562, 1413, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴⁹, 2216, F₃, 31) (dual of [2216, 2067, 32]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(3¹⁴¹, 2188, F₃, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(3¹¹³, 2188, F₃, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (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]

(149−31, 149, 163792)-Net in Base 3 — Upper bound on s

There is no (118, 149, 163793)-net in base 3, because

1 times m-reduction [i] would yield (118, 148, 163793)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41110 904894 579453 042144 271914 133926 267338 654608 913035 648111 522988 111275 > 3¹⁴⁸ [i]

Best Known (149−31, 149, s)-Nets in Base 3

(149−31, 149, 640)-Net over F3 — Constructive and digital

(149−31, 149, 1562)-Net over F3 — Digital

(149−31, 149, 163792)-Net in Base 3 — Upper bound on s

(149−31, 149, 640)-Net over F₃ — Constructive and digital

(149−31, 149, 1562)-Net over F₃ — Digital