MinT - Best Known (168−30, 168, s)-Nets in Base 8

Best Known (168−30, 168, s)-Nets in Base 8

(168−30, 168, 17479)-Net over F₈ — Constructive and digital

Digital (138, 168, 17479)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹⁶⁸, 17479, F₈, 30, 30) (dual of [(17479, 30), 524202, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(8¹⁶⁸, 262185, F₈, 30) (dual of [262185, 262017, 31]-code), using
  - discarding factors / shortening the dual code based on linear OA(8¹⁶⁸, 262191, F₈, 30) (dual of [262191, 262023, 31]-code), using
    - constructionÂ X applied to C_e(29) âŠ‚ C_e(22) [i] based on
      1. linear OA(8¹⁵⁷, 262144, F₈, 30) (dual of [262144, 261987, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
      2. linear OA(8¹²¹, 262144, F₈, 23) (dual of [262144, 262023, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. linear OA(8¹¹, 47, F₈, 6) (dual of [47, 36, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹¹, 63, F₈, 6) (dual of [63, 52, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 8²âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(168−30, 168, 284276)-Net over F₈ — Digital

Digital (138, 168, 284276)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−30, 168, large)-Net in Base 8 — Upper bound on s

There is no (138, 168, large)-net in base 8, because

28 times m-reduction [i] would yield (138, 140, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]