MinT - Best Known (103−11, 103, s)-Nets in Base 7

Best Known (103−11, 103, s)-Nets in Base 7

(103−11, 103, 3355440)-Net over F₇ — Constructive and digital

Digital (92, 103, 3355440)-net over F₇, using

7¹ 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]

(103−11, 103, large)-Net over F₇ — Digital

Digital (92, 103, large)-net over F₇, using

3 times m-reduction [i] based on digital (92, 106, large)-net over F₇, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁰⁶, large, F₇, 14) (dual of [large, large−106, 15]-code), using
  - trace code [i] based on linear OA(49⁵³, 5764801, F₄₉, 14) (dual of [5764801, 5764748, 15]-code), using
    - an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(103−11, 103, large)-Net in Base 7 — Upper bound on s

There is no (92, 103, large)-net in base 7, because

9 times m-reduction [i] would yield (92, 94, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]