MinT - Best Known (138, 166, s)-Nets in Base 8

Best Known (138, 166, s)-Nets in Base 8

(138, 166, 37450)-Net over F₈ — Constructive and digital

Digital (138, 166, 37450)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹⁶⁶, 37450, F₈, 28, 28) (dual of [(37450, 28), 1048434, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(8¹⁶⁶, 524300, F₈, 28) (dual of [524300, 524134, 29]-code), using
  - discarding factors / shortening the dual code based on linear OA(8¹⁶⁶, 524302, F₈, 28) (dual of [524302, 524136, 29]-code), using
    - trace code [i] based on linear OA(64⁸³, 262151, F₆₄, 28) (dual of [262151, 262068, 29]-code), using
      - constructionÂ X applied to C_e(27) âŠ‚ C_e(25) [i] based on
        linear OA(64⁸², 262144, F₆₄, 28) (dual of [262144, 262062, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
        linear OA(64⁷⁶, 262144, F₆₄, 26) (dual of [262144, 262068, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [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]

(138, 166, 556737)-Net over F₈ — Digital

Digital (138, 166, 556737)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 166, large)-Net in Base 8 — Upper bound on s

There is no (138, 166, large)-net in base 8, because

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