MinT - Best Known (109−11, 109, s)-Nets in Base 7

Best Known (109−11, 109, s)-Nets in Base 7

(109−11, 109, 3355461)-Net over F₇ — Constructive and digital

Digital (98, 109, 3355461)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 21)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁷, 21, F₇, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(7⁷, 43, F₇, 5) (dual of [43, 36, 6]-code), using
      - a code by Danev and Olsson [i]
2. digital (91, 102, 3355440)-net over F₇, using
  - trace code for nets [i] based on digital (40, 51, 1677720)-net over F₄₉, using
    - net defined by OOA [i] based on linear OOA(49⁵¹, 1677720, F₄₉, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(49⁵¹, 8388601, F₄₉, 11) (dual of [8388601, 8388550, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁵¹, large, F₄₉, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 28247525Â | 49¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(109−11, 109, large)-Net over F₇ — Digital

Digital (98, 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]

(109−11, 109, large)-Net in Base 7 — Upper bound on s

There is no (98, 109, large)-net in base 7, because

9 times m-reduction [i] would yield (98, 100, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]