MinT - Best Known (105−11, 105, s)-Nets in Base 7

Best Known (105−11, 105, s)-Nets in Base 7

(105−11, 105, 3355440)-Net over F₇ — Constructive and digital

Digital (94, 105, 3355440)-net over F₇, using

7³ times duplication [i] based on 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]

(105−11, 105, large)-Net over F₇ — Digital

Digital (94, 105, large)-net over F₇, using

4 times m-reduction [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]

(105−11, 105, large)-Net in Base 7 — Upper bound on s

There is no (94, 105, large)-net in base 7, because

9 times m-reduction [i] would yield (94, 96, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]