MinT - Best Known (72−8, 72, s)-Nets in Base 8

Best Known (72−8, 72, s)-Nets in Base 8

(72−8, 72, 4198398)-Net over F₈ — Constructive and digital

Digital (64, 72, 4198398)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 14, 4098)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁴, 4098, F₈, 4, 4) (dual of [(4098, 4), 16378, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(8¹⁴, 4098, F₈, 3, 4) (dual of [(4098, 3), 12280, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(8¹⁴, 8196, F₈, 4) (dual of [8196, 8182, 5]-code), using
        trace code [i] based on linear OA(64⁷, 4098, F₆₄, 4) (dual of [4098, 4091, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(64⁷, 4096, F₆₄, 4) (dual of [4096, 4089, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁵, 4096, F₆₄, 3) (dual of [4096, 4091, 4]-code or 4096-cap in PG(4,64)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 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 (50, 58, 4194300)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵⁸, 4194300, F₈, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁵⁸, 8388601, F₈, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁵⁸, 8388602, F₈, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
        trace code [i] based on linear OOA(64²⁹, 4194301, F₆₄, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64²⁹, 8388602, F₆₄, 8) (dual of [8388602, 8388573, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(64²⁹, large, F₆₄, 8) (dual of [large, large−29, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(72−8, 72, 4210556)-Net in Base 8 — Constructive

(64, 72, 4210556)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (10, 14, 16256)-net in base 8, using
  - trace code for nets [i] based on (3, 7, 8128)-net in base 64, using
    - base change [i] based on digital (2, 6, 8128)-net over F₁₂₈, using
      - net defined by OOA [i] based on linear OOA(128⁶, 8128, F₁₂₈, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(128⁶, 16256, F₁₂₈, 4) (dual of [16256, 16250, 5]-code), using
        1 times truncation [i] based on linear OA(128⁷, 16257, F₁₂₈, 5) (dual of [16257, 16250, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (50, 58, 4194300)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8⁵⁸, 4194300, F₈, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(8⁵⁸, 8388601, F₈, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁵⁸, 8388602, F₈, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
        trace code [i] based on linear OOA(64²⁹, 4194301, F₆₄, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64²⁹, 8388602, F₆₄, 8) (dual of [8388602, 8388573, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(64²⁹, large, F₆₄, 8) (dual of [large, large−29, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(72−8, 72, large)-Net over F₈ — Digital

Digital (64, 72, large)-net over F₈, using

t-expansion [i] based on digital (62, 72, large)-net over F₈, using
- 1 times m-reduction [i] based on digital (62, 73, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷³, large, F₈, 11) (dual of [large, large−73, 12]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(72−8, 72, large)-Net in Base 8 — Upper bound on s

There is no (64, 72, large)-net in base 8, because

6 times m-reduction [i] would yield (64, 66, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (72−8, 72, s)-Nets in Base 8

(72−8, 72, 4198398)-Net over F8 — Constructive and digital

(72−8, 72, 4210556)-Net in Base 8 — Constructive

(72−8, 72, large)-Net over F8 — Digital

(72−8, 72, large)-Net in Base 8 — Upper bound on s

(72−8, 72, 4198398)-Net over F₈ — Constructive and digital

(72−8, 72, large)-Net over F₈ — Digital