MinT - Best Known (95−12, 95, s)-Nets in Base 8

Best Known (95−12, 95, s)-Nets in Base 8

(95−12, 95, 2796201)-Net over F₈ — Constructive and digital

Digital (83, 95, 2796201)-net over F₈, using

8² times duplication [i] based on digital (81, 93, 2796201)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁹³, 2796201, F₈, 15, 12) (dual of [(2796201, 15), 41942922, 13]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁹³, 5592403, F₈, 3, 12) (dual of [(5592403, 3), 16777116, 13]-NRT-code), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(8⁹⁰, 5592402, F₈, 3, 12) (dual of [(5592402, 3), 16777116, 13]-NRT-code), using
      - trace code [i] based on linear OOA(64⁴⁵, 2796201, F₆₄, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(95−12, 95, large)-Net over F₈ — Digital

Digital (83, 95, large)-net over F₈, using

2 times m-reduction [i] based on digital (83, 97, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁷, large, F₈, 14) (dual of [large, large−97, 15]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(95−12, 95, large)-Net in Base 8 — Upper bound on s

There is no (83, 95, large)-net in base 8, because

10 times m-reduction [i] would yield (83, 85, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]