MinT - Best Known (110−14, 110, s)-Nets in Base 8

Best Known (110−14, 110, s)-Nets in Base 8

(110−14, 110, 2396742)-Net over F₈ — Constructive and digital

Digital (96, 110, 2396742)-net over F₈, using

8⁴ times duplication [i] based on digital (92, 106, 2396742)-net over F₈, using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(110−14, 110, large)-Net over F₈ — Digital

Digital (96, 110, large)-net over F₈, using

2 times m-reduction [i] based on digital (96, 112, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹², large, F₈, 16) (dual of [large, large−112, 17]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(110−14, 110, large)-Net in Base 8 — Upper bound on s

There is no (96, 110, large)-net in base 8, because

12 times m-reduction [i] would yield (96, 98, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]