MinT - Best Known (165−21, 165, s)-Nets in Base 8

Best Known (165−21, 165, s)-Nets in Base 8

(165−21, 165, 1677720)-Net over F₈ — Constructive and digital

Digital (144, 165, 1677720)-net over F₈, using

8³ times duplication [i] based on digital (141, 162, 1677720)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹⁶², 1677720, F₈, 22, 21) (dual of [(1677720, 22), 36909678, 22]-NRT-code), using
  - OOA 5-folding and stacking with additional row [i] based on linear OOA(8¹⁶², 8388601, F₈, 2, 21) (dual of [(8388601, 2), 16777040, 22]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(8¹⁶², 8388602, F₈, 2, 21) (dual of [(8388602, 2), 16777042, 22]-NRT-code), using
      - trace code [i] based on linear OOA(64⁸¹, 4194301, F₆₄, 2, 21) (dual of [(4194301, 2), 8388521, 22]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁸¹, 8388602, F₆₄, 21) (dual of [8388602, 8388521, 22]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁸¹, large, F₆₄, 21) (dual of [large, large−81, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(165−21, 165, large)-Net over F₈ — Digital

Digital (144, 165, large)-net over F₈, using

3 times m-reduction [i] based on digital (144, 168, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁶⁸, large, F₈, 24) (dual of [large, large−168, 25]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(165−21, 165, large)-Net in Base 8 — Upper bound on s

There is no (144, 165, large)-net in base 8, because

19 times m-reduction [i] would yield (144, 146, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]