MinT - Best Known (90−15, 90, s)-Nets in Base 8

Best Known (90−15, 90, s)-Nets in Base 8

(90−15, 90, 74901)-Net over F₈ — Constructive and digital

Digital (75, 90, 74901)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁹⁰, 74901, F₈, 15, 15) (dual of [(74901, 15), 1123425, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8⁹⁰, 524308, F₈, 15) (dual of [524308, 524218, 16]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁹⁰, 524310, F₈, 15) (dual of [524310, 524220, 16]-code), using
    - trace code [i] based on linear OA(64⁴⁵, 262155, F₆₄, 15) (dual of [262155, 262110, 16]-code), using
      - constructionÂ X applied to C_e(14) âŠ‚ C_e(11) [i] based on
        linear OA(64⁴³, 262144, F₆₄, 15) (dual of [262144, 262101, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(64³⁴, 262144, F₆₄, 12) (dual of [262144, 262110, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(64², 11, F₆₄, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(90−15, 90, 552006)-Net over F₈ — Digital

Digital (75, 90, 552006)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−15, 90, large)-Net in Base 8 — Upper bound on s

There is no (75, 90, large)-net in base 8, because

13 times m-reduction [i] would yield (75, 77, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]