MinT - Best Known (88, 88+12, s)-Nets in Base 7

Best Known (88, 88+12, s)-Nets in Base 7

(88, 88+12, 1921622)-Net over F₇ — Constructive and digital

Digital (88, 100, 1921622)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 10, 21)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 8)-net over F₇, using
      - net from sequence [i] based on digital (0, 7)-sequence over F₇, using
        Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
        the rational function field F₇(x) [i]
        Niederreiter sequence [i]
    2. digital (1, 7, 13)-net over F₇, using
      - 6 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [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]

(88, 88+12, large)-Net over F₇ — Digital

Digital (88, 100, large)-net over F₇, using

7² times duplication [i] based on digital (86, 98, large)-net over F₇, using
- t-expansion [i] based on digital (85, 98, large)-net over F₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁹⁸, large, F₇, 13) (dual of [large, large−98, 14]-code), using
    - trace code [i] based on linear OA(49⁴⁹, 5764802, F₄₉, 13) (dual of [5764802, 5764753, 14]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 5764802Â | 49⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(88, 88+12, large)-Net in Base 7 — Upper bound on s

There is no (88, 100, large)-net in base 7, because

10 times m-reduction [i] would yield (88, 90, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]

Best Known (88, 88+12, s)-Nets in Base 7

(88, 88+12, 1921622)-Net over F7 — Constructive and digital

(88, 88+12, large)-Net over F7 — Digital

(88, 88+12, large)-Net in Base 7 — Upper bound on s

(88, 88+12, 1921622)-Net over F₇ — Constructive and digital

(88, 88+12, large)-Net over F₇ — Digital