MinT - Best Known (146−18, 146, s)-Nets in Base 8

Best Known (146−18, 146, s)-Nets in Base 8

(146−18, 146, 1864134)-Net over F₈ — Constructive and digital

Digital (128, 146, 1864134)-net over F₈, using

8⁵ times duplication [i] based on digital (123, 141, 1864134)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹⁴¹, 1864134, F₈, 21, 18) (dual of [(1864134, 21), 39146673, 19]-NRT-code), using
  - OOA 3-folding and stacking with additional row [i] based on linear OOA(8¹⁴¹, 5592403, F₈, 3, 18) (dual of [(5592403, 3), 16777068, 19]-NRT-code), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(8¹³⁸, 5592402, F₈, 3, 18) (dual of [(5592402, 3), 16777068, 19]-NRT-code), using
      - trace code [i] based on linear OOA(64⁶⁹, 2796201, F₆₄, 3, 18) (dual of [(2796201, 3), 8388534, 19]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64⁶⁹, large, F₆₄, 18) (dual of [large, large−69, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(146−18, 146, large)-Net over F₈ — Digital

Digital (128, 146, large)-net over F₈, using

t-expansion [i] based on digital (125, 146, large)-net over F₈, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁴⁶, large, F₈, 21) (dual of [large, large−146, 22]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8¹⁴⁵, large, F₈, 21) (dual of [large, large−145, 22]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 8¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(146−18, 146, large)-Net in Base 8 — Upper bound on s

There is no (128, 146, large)-net in base 8, because

16 times m-reduction [i] would yield (128, 130, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]