MinT - Best Known (45−9, 45, s)-Nets in Base 64

Best Known (45−9, 45, s)-Nets in Base 64

(45−9, 45, 2228288)-Net over F₆₄ — Constructive and digital

Digital (36, 45, 2228288)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (8, 12, 131138)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64¹², 131138, F₆₄, 4, 4) (dual of [(131138, 4), 524540, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(64¹², 131138, F₆₄, 3, 4) (dual of [(131138, 3), 393402, 5]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        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(64¹⁰, 131073, F₆₄, 3, 4) (dual of [(131073, 3), 393209, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(64¹⁰, 262146, F₆₄, 4) (dual of [262146, 262136, 5]-code), using
        discarding factors / shortening the dual code 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 (24, 33, 2097150)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³³, 2097150, F₆₄, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(64³³, 8388601, F₆₄, 9) (dual of [8388601, 8388568, 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]

(45−9, 45, 3145727)-Net in Base 64 — Constructive

(36, 45, 3145727)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (8, 12, 1048577)-net in base 64, using
  - net defined by OOA [i] based on OOA(64¹², 1048577, S₆₄, 4, 4), using
    - appending kth column [i] based on OOA(64¹², 1048577, S₆₄, 3, 4), using
      - OA 2-folding and stacking [i] based on OA(64¹², 2097154, S₆₄, 4), using
        discarding factors based on OA(64¹², 2097155, S₆₄, 4), using
        discarding parts of the base [i] based on linear OA(128¹⁰, 2097155, F₁₂₈, 4) (dual of [2097155, 2097145, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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, 2097152, F₁₂₈, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 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 (24, 33, 2097150)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³³, 2097150, F₆₄, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(64³³, 8388601, F₆₄, 9) (dual of [8388601, 8388568, 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]

(45−9, 45, large)-Net over F₆₄ — Digital

Digital (36, 45, large)-net over F₆₄, using

t-expansion [i] based on digital (34, 45, large)-net over F₆₄, using
- 1 times m-reduction [i] based on digital (34, 46, large)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁴⁶, large, F₆₄, 12) (dual of [large, large−46, 13]-code), using
    - 1 times code embedding in larger space [i] 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]

(45−9, 45, large)-Net in Base 64 — Upper bound on s

There is no (36, 45, large)-net in base 64, because

7 times m-reduction [i] would yield (36, 38, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (45−9, 45, s)-Nets in Base 64

(45−9, 45, 2228288)-Net over F64 — Constructive and digital

(45−9, 45, 3145727)-Net in Base 64 — Constructive

(45−9, 45, large)-Net over F64 — Digital

(45−9, 45, large)-Net in Base 64 — Upper bound on s

(45−9, 45, 2228288)-Net over F₆₄ — Constructive and digital

(45−9, 45, large)-Net over F₆₄ — Digital