MinT - Best Known (72−10, 72, s)-Nets in Base 8

Best Known (72−10, 72, s)-Nets in Base 8

(72−10, 72, 1677748)-Net over F₈ — Constructive and digital

Digital (62, 72, 1677748)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 28)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁷, 28, F₈, 5, 5) (dual of [(28, 5), 133, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
      - a code by Danev and Olsson [i]
2. digital (55, 65, 1677720)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁶⁵, 1677720, F₈, 10, 10) (dual of [(1677720, 10), 16777135, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(8⁶⁵, 8388600, F₈, 10) (dual of [8388600, 8388535, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁶⁵, large, F₈, 10) (dual of [large, large−65, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(72−10, 72, large)-Net over F₈ — Digital

Digital (62, 72, large)-net over F₈, using

1 times m-reduction [i] based on digital (62, 73, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷³, large, F₈, 11) (dual of [large, large−73, 12]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(72−10, 72, large)-Net in Base 8 — Upper bound on s

There is no (62, 72, large)-net in base 8, because

8 times m-reduction [i] would yield (62, 64, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]