MinT - Best Known (150−19, 150, s)-Nets in Base 8

Best Known (150−19, 150, s)-Nets in Base 8

(150−19, 150, 1864134)-Net over F₈ — Constructive and digital

Digital (131, 150, 1864134)-net over F₈, using

8¹ times duplication [i] based on digital (130, 149, 1864134)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹⁴⁹, 1864134, F₈, 21, 19) (dual of [(1864134, 21), 39146665, 20]-NRT-code), using
  - OOA 3-folding and stacking with additional row [i] based on linear OOA(8¹⁴⁹, 5592403, F₈, 3, 19) (dual of [(5592403, 3), 16777060, 20]-NRT-code), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(8¹⁴⁶, 5592402, F₈, 3, 19) (dual of [(5592402, 3), 16777060, 20]-NRT-code), using
      - trace code [i] based on linear OOA(64⁷³, 2796201, F₆₄, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64⁷³, large, F₆₄, 19) (dual of [large, large−73, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(150−19, 150, large)-Net over F₈ — Digital

Digital (131, 150, large)-net over F₈, using

3 times m-reduction [i] based on digital (131, 153, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁵³, large, F₈, 22) (dual of [large, large−153, 23]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(150−19, 150, large)-Net in Base 8 — Upper bound on s

There is no (131, 150, large)-net in base 8, because

17 times m-reduction [i] would yield (131, 133, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]