MinT - Best Known (65−12, 65, s)-Nets in Base 64

Best Known (65−12, 65, s)-Nets in Base 64

(65−12, 65, 1489579)-Net over F₆₄ — Constructive and digital

Digital (53, 65, 1489579)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (14, 20, 91479)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 4097)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64⁴, 4097, F₆₄, 3, 3) (dual of [(4097, 3), 12287, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(64⁴, 4097, F₆₄, 2, 3) (dual of [(4097, 2), 8190, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁴, 4097, F₆₄, 3) (dual of [4097, 4093, 4]-code or 4097-cap in PG(3,64)), using
        ovoid in PG(3,Â 64) [i]
    2. digital (10, 16, 87382)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64¹⁶, 87382, F₆₄, 6, 6) (dual of [(87382, 6), 524276, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(64¹⁶, 262146, F₆₄, 6) (dual of [262146, 262130, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹⁶, 262147, F₆₄, 6) (dual of [262147, 262131, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(64¹⁶, 262144, F₆₄, 6) (dual of [262144, 262128, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(64¹³, 262144, F₆₄, 5) (dual of [262144, 262131, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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 (33, 45, 1398100)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁵, 1398100, F₆₄, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(64⁴⁵, 8388600, F₆₄, 12) (dual of [8388600, 8388555, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(65−12, 65, 2097153)-Net in Base 64 — Constructive

(53, 65, 2097153)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (14, 20, 699053)-net in base 64, using
  - net defined by OOA [i] based on OOA(64²⁰, 699053, S₆₄, 6, 6), using
    - OA 3-folding and stacking [i] based on OA(64²⁰, 2097159, S₆₄, 6), using
      - discarding factors based on OA(64²⁰, 2097160, S₆₄, 6), using
        discarding parts of the base [i] based on linear OA(128¹⁷, 2097160, F₁₂₈, 6) (dual of [2097160, 2097143, 7]-code), using
        constructionÂ X4 applied to C_e(5) âŠ‚ C_e(3) [i] based on
        linear OA(128¹⁶, 2097152, F₁₂₈, 6) (dual of [2097152, 2097136, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(128¹⁰, 2097152, F₁₂₈, 4) (dual of [2097152, 2097142, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(128⁷, 8, F₁₂₈, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
        dual of repetition code with length 8 [i]
        linear OA(128¹, 8, F₁₂₈, 1) (dual of [8, 7, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, 128, F₁₂₈, 1) (dual of [128, 127, 2]-code), using
        Reedâ€“Solomon code RS(127,128) [i]
2. digital (33, 45, 1398100)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁵, 1398100, F₆₄, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(64⁴⁵, 8388600, F₆₄, 12) (dual of [8388600, 8388555, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(65−12, 65, large)-Net over F₆₄ — Digital

Digital (53, 65, large)-net over F₆₄, using

6 times m-reduction [i] based on digital (53, 71, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁷¹, large, F₆₄, 18) (dual of [large, large−71, 19]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(64⁶⁹, large, F₆₄, 18) (dual of [large, large−69, 19]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(65−12, 65, large)-Net in Base 64 — Upper bound on s

There is no (53, 65, large)-net in base 64, because

10 times m-reduction [i] would yield (53, 55, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (65−12, 65, s)-Nets in Base 64

(65−12, 65, 1489579)-Net over F64 — Constructive and digital

(65−12, 65, 2097153)-Net in Base 64 — Constructive

(65−12, 65, large)-Net over F64 — Digital

(65−12, 65, large)-Net in Base 64 — Upper bound on s

(65−12, 65, 1489579)-Net over F₆₄ — Constructive and digital

(65−12, 65, large)-Net over F₆₄ — Digital