MinT - Best Known (87−11, 87, s)-Nets in Base 7

Best Known (87−11, 87, s)-Nets in Base 7

(87−11, 87, 2305928)-Net over F₇ — Constructive and digital

Digital (76, 87, 2305928)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 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 (71, 82, 2305920)-net over F₇, using
  - trace code for nets [i] based on digital (30, 41, 1152960)-net over F₄₉, using
    - net defined by OOA [i] based on linear OOA(49⁴¹, 1152960, F₄₉, 11, 11) (dual of [(1152960, 11), 12682519, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on 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]

(87−11, 87, large)-Net over F₇ — Digital

Digital (76, 87, large)-net over F₇, using

7⁵ times duplication [i] based on digital (71, 82, large)-net over F₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁸², large, F₇, 11) (dual of [large, large−82, 12]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 20176803Â | 7⁹âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(87−11, 87, large)-Net in Base 7 — Upper bound on s

There is no (76, 87, large)-net in base 7, because

9 times m-reduction [i] would yield (76, 78, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]