MinT - Best Known (57, 57+15, s)-Nets in Base 7

Best Known (57, 57+15, s)-Nets in Base 7

(57, 57+15, 2422)-Net over F₇ — Constructive and digital

Digital (57, 72, 2422)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 11, 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, 8, 13)-net over F₇, using
      - 5 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
2. digital (46, 61, 2401)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁶¹, 2401, F₇, 15, 15) (dual of [(2401, 15), 35954, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(7⁶¹, 16808, F₇, 15) (dual of [16808, 16747, 16]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16808Â | 7¹⁰âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(57, 57+15, 22370)-Net over F₇ — Digital

Digital (57, 72, 22370)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+15, large)-Net in Base 7 — Upper bound on s

There is no (57, 72, large)-net in base 7, because

13 times m-reduction [i] would yield (57, 59, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]