MinT - Best Known (78−10, 78, s)-Nets in Base 8

Best Known (78−10, 78, s)-Nets in Base 8

(78−10, 78, 3355440)-Net over F₈ — Constructive and digital

Digital (68, 78, 3355440)-net over F₈, using

8⁴ times duplication [i] based on digital (64, 74, 3355440)-net over F₈, using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³⁷, 1677720, F₆₄, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(64³⁷, 8388600, F₆₄, 10) (dual of [8388600, 8388563, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(78−10, 78, large)-Net over F₈ — Digital

Digital (68, 78, large)-net over F₈, using

8⁵ times duplication [i] based on digital (63, 73, large)-net over F₈, using
- t-expansion [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]

(78−10, 78, large)-Net in Base 8 — Upper bound on s

There is no (68, 78, large)-net in base 8, because

8 times m-reduction [i] would yield (68, 70, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]