MinT - Best Known (143−27, 143, s)-Nets in Base 9

Best Known (143−27, 143, s)-Nets in Base 9

(143−27, 143, 4586)-Net over F₉ — Constructive and digital

Digital (116, 143, 4586)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (10, 23, 44)-net over F₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 7, 16)-net over F₉, using
      - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
        a function field by GarcÃa and Quoos [i]
    2. digital (3, 16, 28)-net over F₉, using
      - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
        the Hermitian function field over F₉ [i]
2. digital (93, 120, 4542)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹²⁰, 4542, F₉, 27, 27) (dual of [(4542, 27), 122514, 28]-NRT-code), using
    - OOA 13-folding and stacking with additional row [i] based on linear OA(9¹²⁰, 59047, F₉, 27) (dual of [59047, 58927, 28]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹²⁰, 59048, F₉, 27) (dual of [59048, 58928, 28]-code), using
        1 times truncation [i] based on linear OA(9¹²¹, 59049, F₉, 28) (dual of [59049, 58928, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(143−27, 143, 4590)-Net in Base 9 — Constructive

(116, 143, 4590)-net in base 9, using

(u,Â u+v)-construction [i] based on
1. (10, 23, 48)-net in base 9, using
  - 1 times m-reduction [i] based on (10, 24, 48)-net in base 9, using
    - base change [i] based on digital (2, 16, 48)-net over F₂₇, using
      - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
        a function field by GarcÃa and Quoos [i]
2. digital (93, 120, 4542)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹²⁰, 4542, F₉, 27, 27) (dual of [(4542, 27), 122514, 28]-NRT-code), using
    - OOA 13-folding and stacking with additional row [i] based on linear OA(9¹²⁰, 59047, F₉, 27) (dual of [59047, 58927, 28]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹²⁰, 59048, F₉, 27) (dual of [59048, 58928, 28]-code), using
        1 times truncation [i] based on linear OA(9¹²¹, 59049, F₉, 28) (dual of [59049, 58928, 29]-code), using
        an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(143−27, 143, 233647)-Net over F₉ — Digital

Digital (116, 143, 233647)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(143−27, 143, large)-Net in Base 9 — Upper bound on s

There is no (116, 143, large)-net in base 9, because

25 times m-reduction [i] would yield (116, 118, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (143−27, 143, s)-Nets in Base 9

(143−27, 143, 4586)-Net over F9 — Constructive and digital

(143−27, 143, 4590)-Net in Base 9 — Constructive

(143−27, 143, 233647)-Net over F9 — Digital

(143−27, 143, large)-Net in Base 9 — Upper bound on s

(143−27, 143, 4586)-Net over F₉ — Constructive and digital

(143−27, 143, 233647)-Net over F₉ — Digital