MinT - Best Known (49−10, 49, s)-Nets in Base 8

Best Known (49−10, 49, s)-Nets in Base 8

(49−10, 49, 52430)-Net over F₈ — Constructive and digital

Digital (39, 49, 52430)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁴⁹, 52430, F₈, 10, 10) (dual of [(52430, 10), 524251, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8⁴⁹, 262150, F₈, 10) (dual of [262150, 262101, 11]-code), using
  - constructionÂ X applied to C_e(9) âŠ‚ C_e(8) [i] based on
    1. linear OA(8⁴⁹, 262144, F₈, 10) (dual of [262144, 262095, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    2. linear OA(8⁴³, 262144, F₈, 9) (dual of [262144, 262101, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(8⁰, 6, F₈, 0) (dual of [6, 6, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(49−10, 49, 140968)-Net over F₈ — Digital

Digital (39, 49, 140968)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴⁹, 140968, F₈, 10) (dual of [140968, 140919, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁴⁹, 262144, F₈, 10) (dual of [262144, 262095, 11]-code), using
  - an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(49−10, 49, large)-Net in Base 8 — Upper bound on s

There is no (39, 49, large)-net in base 8, because

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