MinT - Best Known (111, 111+14, s)-Nets in Base 9

Best Known (111, 111+14, s)-Nets in Base 9

(111, 111+14, 2397070)-Net over F₉ — Constructive and digital

Digital (111, 125, 2397070)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (12, 19, 328)-net over F₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 5, 212)-net over F₉, using
      - net defined by OOA [i] based on linear OOA(9⁵, 212, F₉, 3, 3) (dual of [(212, 3), 631, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(9⁵, 212, F₉, 2, 3) (dual of [(212, 2), 419, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9⁵, 212, F₉, 3) (dual of [212, 207, 4]-code or 212-cap in PG(4,9)), using
        caps completed using a computer search [i]
    2. digital (7, 14, 164)-net over F₉, using
      - trace code for nets [i] based on digital (0, 7, 82)-net over F₈₁, using
        net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]
2. digital (92, 106, 2396742)-net over F₉, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁵³, 1198371, F₈₁, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(81⁵³, 8388597, F₈₁, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵³, large, F₈₁, 14) (dual of [large, large−53, 15]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(111, 111+14, large)-Net over F₉ — Digital

Digital (111, 125, large)-net over F₉, using

t-expansion [i] based on digital (110, 125, large)-net over F₉, using
- 4 times m-reduction [i] based on digital (110, 129, large)-net over F₉, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹²⁹, large, F₉, 19) (dual of [large, large−129, 20]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 21523361Â | 9¹⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(111, 111+14, large)-Net in Base 9 — Upper bound on s

There is no (111, 125, large)-net in base 9, because

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

Best Known (111, 111+14, s)-Nets in Base 9

(111, 111+14, 2397070)-Net over F9 — Constructive and digital

(111, 111+14, large)-Net over F9 — Digital

(111, 111+14, large)-Net in Base 9 — Upper bound on s

(111, 111+14, 2397070)-Net over F₉ — Constructive and digital

(111, 111+14, large)-Net over F₉ — Digital