MinT - Best Known (111, 140, s)-Nets in Base 9

Best Known (111, 140, s)-Nets in Base 9

(111, 140, 4228)-Net over F₉ — Constructive and digital

Digital (111, 140, 4228)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 10)-net over F₉, using
  - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
      - the rational function field F₉(x) [i]
    - Niederreiter sequence [i]
2. digital (97, 126, 4218)-net over F₉, using
  - net defined by OOA [i] based on linear OOA(9¹²⁶, 4218, F₉, 29, 29) (dual of [(4218, 29), 122196, 30]-NRT-code), using
    - OOA 14-folding and stacking with additional row [i] based on linear OA(9¹²⁶, 59053, F₉, 29) (dual of [59053, 58927, 30]-code), using
      - discarding factors / shortening the dual code based on linear OA(9¹²⁶, 59054, F₉, 29) (dual of [59054, 58928, 30]-code), using
        constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
        linear OA(9¹²⁶, 59049, F₉, 29) (dual of [59049, 58923, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 9⁵âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
        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]
        linear OA(9⁰, 5, F₉, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(111, 140, 83406)-Net over F₉ — Digital

Digital (111, 140, 83406)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 140, large)-Net in Base 9 — Upper bound on s

There is no (111, 140, large)-net in base 9, because

27 times m-reduction [i] would yield (111, 113, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (111, 140, s)-Nets in Base 9

(111, 140, 4228)-Net over F9 — Constructive and digital

(111, 140, 83406)-Net over F9 — Digital

(111, 140, large)-Net in Base 9 — Upper bound on s

(111, 140, 4228)-Net over F₉ — Constructive and digital

(111, 140, 83406)-Net over F₉ — Digital