MinT - Best Known (65, 65+8, s)-Nets in Base 8

Best Known (65, 65+8, s)-Nets in Base 8

(65, 65+8, 4198460)-Net over F₈ — Constructive and digital

Digital (65, 73, 4198460)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 15, 4160)-net over F₈, using
  - base reduction for projective spaces (embedding PG(7,64) in PG(14,8)) for nets [i] based on digital (4, 8, 4160)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁸, 4160, F₆₄, 4, 4) (dual of [(4160, 4), 16632, 5]-NRT-code), using
      - appending kth column [i] based on linear OOA(64⁸, 4160, F₆₄, 3, 4) (dual of [(4160, 3), 12472, 5]-NRT-code), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(64⁰, s, F₆₄, 3, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64¹, 65, F₆₄, 3, 1) (dual of [(65, 3), 194, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(64¹, s, F₆₄, 3, 1) with arbitrarily large s, using
        appending 2 arbitrary columns [i] based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(64¹, 65, F₆₄, 3, 1) (dual of [(65, 3), 194, 2]-NRT-code) (see above)
        linear OOA(64², 65, F₆₄, 3, 2) (dual of [(65, 3), 193, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;193,64) [i]
        linear OOA(64⁴, 65, F₆₄, 3, 4) (dual of [(65, 3), 191, 5]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;191,64) [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]

(65, 65+8, 4226940)-Net in Base 8 — Constructive

(65, 73, 4226940)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (11, 15, 32640)-net in base 8, using
  - net defined by OOA [i] based on OOA(8¹⁵, 32640, S₈, 4, 4), using
    - appending kth column [i] based on OOA(8¹⁵, 32640, S₈, 3, 4), using
      - OA 2-folding and stacking [i] based on OA(8¹⁵, 65280, S₈, 4), using
        discarding parts of the base [i] based on linear OA(16¹¹, 65280, F₁₆, 4) (dual of [65280, 65269, 5]-code), using
        base reduction for projective spaces (embedding PG(5,256) in PG(10,16)) [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
        1 times truncation [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 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]

(65, 65+8, large)-Net over F₈ — Digital

Digital (65, 73, large)-net over F₈, using

t-expansion [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]

(65, 65+8, large)-Net in Base 8 — Upper bound on s

There is no (65, 73, large)-net in base 8, because

6 times m-reduction [i] would yield (65, 67, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (65, 65+8, s)-Nets in Base 8

(65, 65+8, 4198460)-Net over F8 — Constructive and digital

(65, 65+8, 4226940)-Net in Base 8 — Constructive

(65, 65+8, large)-Net over F8 — Digital

(65, 65+8, large)-Net in Base 8 — Upper bound on s

(65, 65+8, 4198460)-Net over F₈ — Constructive and digital

(65, 65+8, large)-Net over F₈ — Digital