MinT - Best Known (172−33, 172, s)-Nets in Base 8

Best Known (172−33, 172, s)-Nets in Base 8

(172−33, 172, 16384)-Net over F₈ — Constructive and digital

Digital (139, 172, 16384)-net over F₈, using

8³ times duplication [i] based on digital (136, 169, 16384)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹⁶⁹, 16384, F₈, 33, 33) (dual of [(16384, 33), 540503, 34]-NRT-code), using
  - OOA 16-folding and stacking with additional row [i] based on linear OA(8¹⁶⁹, 262145, F₈, 33) (dual of [262145, 261976, 34]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 262145Â | 8¹²âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(172−33, 172, 169999)-Net over F₈ — Digital

Digital (139, 172, 169999)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁷², 169999, F₈, 33) (dual of [169999, 169827, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹⁷², 262155, F₈, 33) (dual of [262155, 261983, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,14]) [i] based on
    1. linear OA(8¹⁶⁹, 262145, F₈, 33) (dual of [262145, 261976, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 8¹²âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(8¹⁵⁷, 262145, F₈, 29) (dual of [262145, 261988, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 8¹²âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    3. linear OA(8³, 10, F₈, 3) (dual of [10, 7, 4]-code or 10-arc in PG(2,8) or 10-cap in PG(2,8)), using
      - hyperoval in PG(2,Â 8) [i]

(172−33, 172, large)-Net in Base 8 — Upper bound on s

There is no (139, 172, large)-net in base 8, because

31 times m-reduction [i] would yield (139, 141, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]