MinT - Best Known (109−12, 109, s)-Nets in Base 7

Best Known (109−12, 109, s)-Nets in Base 7

(109−12, 109, 1921809)-Net over F₇ — Constructive and digital

Digital (97, 109, 1921809)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (13, 19, 208)-net over F₇, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 8)-net over F₇, using
      - kÂ =Â 2 construction [i]
    2. digital (2, 5, 100)-net over F₇, using
      - s-reduction based on digital (2, 5, 132)-net over F₇, using
        net defined by OOA [i] based on linear OOA(7⁵, 132, F₇, 3, 3) (dual of [(132, 3), 391, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(7⁵, 132, F₇, 2, 3) (dual of [(132, 2), 259, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(7⁵, 132, F₇, 3) (dual of [132, 127, 4]-code or 132-cap in PG(4,7)), using
        caps completed using a computer search [i]
    3. digital (6, 12, 100)-net over F₇, using
      - trace code for nets [i] based on digital (0, 6, 50)-net over F₄₉, using
        net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
        Niederreiter sequence [i]
2. digital (78, 90, 1921601)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁹⁰, 1921601, F₇, 14, 12) (dual of [(1921601, 14), 26902324, 13]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(7⁹⁰, 5764804, F₇, 2, 12) (dual of [(5764804, 2), 11529518, 13]-NRT-code), using
      - trace code [i] based on linear OOA(49⁴⁵, 2882402, F₄₉, 2, 12) (dual of [(2882402, 2), 5764759, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(49⁴⁵, 5764804, F₄₉, 12) (dual of [5764804, 5764759, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁴⁵, 5764805, F₄₉, 12) (dual of [5764805, 5764760, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(49⁴⁵, 5764801, F₄₉, 12) (dual of [5764801, 5764756, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(49⁴¹, 5764801, F₄₉, 11) (dual of [5764801, 5764760, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(109−12, 109, large)-Net over F₇ — Digital

Digital (97, 109, large)-net over F₇, using

t-expansion [i] based on digital (94, 109, large)-net over F₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁰⁹, large, F₇, 15) (dual of [large, large−109, 16]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 20176804Â | 7¹⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(109−12, 109, large)-Net in Base 7 — Upper bound on s

There is no (97, 109, large)-net in base 7, because

10 times m-reduction [i] would yield (97, 99, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (109−12, 109, s)-Nets in Base 7

(109−12, 109, 1921809)-Net over F7 — Constructive and digital

(109−12, 109, large)-Net over F7 — Digital

(109−12, 109, large)-Net in Base 7 — Upper bound on s

(109−12, 109, 1921809)-Net over F₇ — Constructive and digital

(109−12, 109, large)-Net over F₇ — Digital