MinT - Best Known (169−30, 169, s)-Nets in Base 8

Best Known (169−30, 169, s)-Nets in Base 8

(169−30, 169, 17479)-Net over F₈ — Constructive and digital

Digital (139, 169, 17479)-net over F₈, using

1 times m-reduction [i] based on digital (139, 170, 17479)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹⁷⁰, 17479, F₈, 31, 31) (dual of [(17479, 31), 541679, 32]-NRT-code), using
  - OOA 15-folding and stacking with additional row [i] based on linear OA(8¹⁷⁰, 262186, F₈, 31) (dual of [262186, 262016, 32]-code), using
    - discarding factors / shortening the dual code based on linear OA(8¹⁷⁰, 262187, F₈, 31) (dual of [262187, 262017, 32]-code), using
      - constructionÂ X applied to C_e(30) âŠ‚ C_e(24) [i] based on
        linear OA(8¹⁶³, 262144, F₈, 31) (dual of [262144, 261981, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
        linear OA(8¹²⁷, 262144, F₈, 25) (dual of [262144, 262017, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 8⁶âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        linear OA(8⁷, 43, F₈, 5) (dual of [43, 36, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(8⁷, 57, F₈, 5) (dual of [57, 50, 6]-code), using
        a code by Danev and Olsson [i]

(169−30, 169, 305407)-Net over F₈ — Digital

Digital (139, 169, 305407)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−30, 169, large)-Net in Base 8 — Upper bound on s

There is no (139, 169, large)-net in base 8, because

28 times m-reduction [i] would yield (139, 141, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]