MinT - Best Known (47−8, 47, s)-Nets in Base 8

Best Known (47−8, 47, s)-Nets in Base 8

(47−8, 47, 131075)-Net over F₈ — Constructive and digital

Digital (39, 47, 131075)-net over F₈, using

8¹ times duplication [i] based on digital (38, 46, 131075)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8⁴⁶, 131075, F₈, 8, 8) (dual of [(131075, 8), 1048554, 9]-NRT-code), using
  - OA 4-folding and stacking [i] based on linear OA(8⁴⁶, 524300, F₈, 8) (dual of [524300, 524254, 9]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁴⁶, 524302, F₈, 8) (dual of [524302, 524256, 9]-code), using
      - trace code [i] based on linear OA(64²³, 262151, F₆₄, 8) (dual of [262151, 262128, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(5) [i] based on
        linear OA(64²², 262144, F₆₄, 8) (dual of [262144, 262122, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(64¹⁶, 262144, F₆₄, 6) (dual of [262144, 262128, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(64¹, 7, F₆₄, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(47−8, 47, 559020)-Net over F₈ — Digital

Digital (39, 47, 559020)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(47−8, 47, large)-Net in Base 8 — Upper bound on s

There is no (39, 47, large)-net in base 8, because

6 times m-reduction [i] would yield (39, 41, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]