MinT - Best Known (88, 88+10, s)-Nets in Base 7

Best Known (88, 88+10, s)-Nets in Base 7

(88, 88+10, 3355453)-Net over F₇ — Constructive and digital

Digital (88, 98, 3355453)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 13)-net over F₇, using
  - 7 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (82, 92, 3355440)-net over F₇, using
  - trace code for nets [i] based on digital (36, 46, 1677720)-net over F₄₉, using
    - net defined by OOA [i] based on linear OOA(49⁴⁶, 1677720, F₄₉, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(49⁴⁶, 8388600, F₄₉, 10) (dual of [8388600, 8388554, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁴⁶, large, F₄₉, 10) (dual of [large, large−46, 11]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 25679568Â | 49⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(88, 88+10, large)-Net over F₇ — Digital

Digital (88, 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+10, large)-Net in Base 7 — Upper bound on s

There is no (88, 98, large)-net in base 7, because

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