MinT - Best Known (86−9, 86, s)-Nets in Base 8

Best Known (86−9, 86, s)-Nets in Base 8

(86−9, 86, 4456447)-Net over F₈ — Constructive and digital

Digital (77, 86, 4456447)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (16, 20, 262147)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8²⁰, 262147, F₈, 4, 4) (dual of [(262147, 4), 1048568, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8²⁰, 262147, F₈, 3, 4) (dual of [(262147, 3), 786421, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(8²⁰, 524294, F₈, 4) (dual of [524294, 524274, 5]-code), using
        trace code [i] based on linear OA(64¹⁰, 262147, F₆₄, 4) (dual of [262147, 262137, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁷, 262144, F₆₄, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (57, 66, 4194300)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁶⁶, 4194300, F₈, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁶⁶, 8388601, F₈, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁶⁶, 8388602, F₈, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
        trace code [i] based on linear OOA(64³³, 4194301, F₆₄, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64³³, 8388602, F₆₄, 9) (dual of [8388602, 8388569, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(64³³, large, F₆₄, 9) (dual of [large, large−33, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(86−9, 86, large)-Net over F₈ — Digital

Digital (77, 86, large)-net over F₈, using

t-expansion [i] based on digital (76, 86, large)-net over F₈, using
- 3 times m-reduction [i] based on digital (76, 89, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁹, large, F₈, 13) (dual of [large, large−89, 14]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(86−9, 86, large)-Net in Base 8 — Upper bound on s

There is no (77, 86, large)-net in base 8, because

7 times m-reduction [i] would yield (77, 79, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (86−9, 86, s)-Nets in Base 8

(86−9, 86, 4456447)-Net over F8 — Constructive and digital

(86−9, 86, large)-Net over F8 — Digital

(86−9, 86, large)-Net in Base 8 — Upper bound on s

(86−9, 86, 4456447)-Net over F₈ — Constructive and digital

(86−9, 86, large)-Net over F₈ — Digital