MinT - Best Known (85−11, 85, s)-Nets in Base 8

Best Known (85−11, 85, s)-Nets in Base 8

(85−11, 85, 3355440)-Net over F₈ — Constructive and digital

Digital (74, 85, 3355440)-net over F₈, using

8³ times duplication [i] based on digital (71, 82, 3355440)-net over F₈, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(85−11, 85, large)-Net over F₈ — Digital

Digital (74, 85, large)-net over F₈, using

8⁴ times duplication [i] based on digital (70, 81, large)-net over F₈, using
- t-expansion [i] based on digital (69, 81, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸¹, large, F₈, 12) (dual of [large, large−81, 13]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(85−11, 85, large)-Net in Base 8 — Upper bound on s

There is no (74, 85, large)-net in base 8, because

9 times m-reduction [i] would yield (74, 76, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]