MinT - Best Known (153−27, 153, s)-Nets in Base 3

Best Known (153−27, 153, s)-Nets in Base 3

(153−27, 153, 700)-Net over F₃ — Constructive and digital

Digital (126, 153, 700)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 17, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (109, 136, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 34, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(153−27, 153, 4027)-Net over F₃ — Digital

Digital (126, 153, 4027)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵³, 4027, F₃, 27) (dual of [4027, 3874, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵³, 6590, F₃, 27) (dual of [6590, 6437, 28]-code), using
  - constructionÂ X applied to C([0,13]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(3¹⁴⁵, 6562, F₃, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    2. linear OA(3¹¹³, 6562, F₃, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (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]

(153−27, 153, 1074037)-Net in Base 3 — Upper bound on s

There is no (126, 153, 1074038)-net in base 3, because

1 times m-reduction [i] would yield (126, 152, 1074038)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 329896 599910 793808 078605 618738 675517 506134 072311 987233 344851 385951 454517 > 3¹⁵² [i]

Best Known (153−27, 153, s)-Nets in Base 3

(153−27, 153, 700)-Net over F3 — Constructive and digital

(153−27, 153, 4027)-Net over F3 — Digital

(153−27, 153, 1074037)-Net in Base 3 — Upper bound on s

(153−27, 153, 700)-Net over F₃ — Constructive and digital

(153−27, 153, 4027)-Net over F₃ — Digital