MinT - Best Known (67, 67+21, s)-Nets in Base 8

Best Known (67, 67+21, s)-Nets in Base 8

(67, 67+21, 821)-Net over F₈ — Constructive and digital

Digital (67, 88, 821)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁸⁸, 821, F₈, 21, 21) (dual of [(821, 21), 17153, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8⁸⁸, 8211, F₈, 21) (dual of [8211, 8123, 22]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁸⁸, 8216, F₈, 21) (dual of [8216, 8128, 22]-code), using
    - trace code [i] based on linear OA(64⁴⁴, 4108, F₆₄, 21) (dual of [4108, 4064, 22]-code), using
      - constructionÂ X applied to C([0,10]) âŠ‚ C([0,8]) [i] based on
        linear OA(64⁴¹, 4097, F₆₄, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(64³³, 4097, F₆₄, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(64³, 11, F₆₄, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
        discarding factors / shortening the dual code based on linear OA(64³, 64, F₆₄, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
        Reedâ€“Solomon code RS(61,64) [i]

(67, 67+21, 1032)-Net in Base 8 — Constructive

(67, 88, 1032)-net in base 8, using

base change [i] based on digital (45, 66, 1032)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (12, 22, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 11, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
  2. digital (23, 44, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 22, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(67, 67+21, 11173)-Net over F₈ — Digital

Digital (67, 88, 11173)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+21, large)-Net in Base 8 — Upper bound on s

There is no (67, 88, large)-net in base 8, because

19 times m-reduction [i] would yield (67, 69, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]