MinT - Best Known (162−34, 162, s)-Nets in Base 8

Best Known (162−34, 162, s)-Nets in Base 8

(162−34, 162, 1931)-Net over F₈ — Constructive and digital

Digital (128, 162, 1931)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹⁶², 1931, F₈, 34, 34) (dual of [(1931, 34), 65492, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(8¹⁶², 32827, F₈, 34) (dual of [32827, 32665, 35]-code), using
  - discarding factors / shortening the dual code based on linear OA(8¹⁶², 32829, F₈, 34) (dual of [32829, 32667, 35]-code), using
    - constructionÂ X applied to C_e(33) âŠ‚ C_e(22) [i] based on
      1. linear OA(8¹⁴⁶, 32768, F₈, 34) (dual of [32768, 32622, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      2. linear OA(8¹⁰¹, 32768, F₈, 23) (dual of [32768, 32667, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 8⁵âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. linear OA(8¹⁶, 61, F₈, 10) (dual of [61, 45, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹⁶, 63, F₈, 10) (dual of [63, 47, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 8²âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(162−34, 162, 51021)-Net over F₈ — Digital

Digital (128, 162, 51021)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−34, 162, large)-Net in Base 8 — Upper bound on s

There is no (128, 162, large)-net in base 8, because

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