MinT - Best Known (53−9, 53, s)-Nets in Base 64

Best Known (53−9, 53, s)-Nets in Base 64

(53−9, 53, 6291450)-Net over F₆₄ — Constructive and digital

Digital (44, 53, 6291450)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 7, 2097150)-net over F₆₄, using
  - s-reduction based on digital (4, 7, large)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁷, large, F₆₄, 3, 3), using
      - appending kth column [i] based on linear OOA(64⁷, large, F₆₄, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁷, large, F₆₄, 3) (dual of [large, large−7, 4]-code), using
        product of projective cap with 64²-ovoid cap in AG(3,Â 64) [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 (9, 13, 2097150)-net over F₆₄, using
  - s-reduction based on digital (9, 13, 4194301)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64¹³, 4194301, F₆₄, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
      - appending kth column [i] based on linear OOA(64¹³, 4194301, F₆₄, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(64¹³, 8388602, F₆₄, 4) (dual of [8388602, 8388589, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹³, large, F₆₄, 4) (dual of [large, large−13, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
3. 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]

(53−9, 53, large)-Net over F₆₄ — Digital

Digital (44, 53, large)-net over F₆₄, using

6 times m-reduction [i] based on digital (44, 59, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁹, large, F₆₄, 15) (dual of [large, large−59, 16]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(64⁵⁷, large, F₆₄, 15) (dual of [large, large−57, 16]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(53−9, 53, large)-Net in Base 64 — Upper bound on s

There is no (44, 53, large)-net in base 64, because

7 times m-reduction [i] would yield (44, 46, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (53−9, 53, s)-Nets in Base 64

(53−9, 53, 6291450)-Net over F64 — Constructive and digital

(53−9, 53, large)-Net over F64 — Digital

(53−9, 53, large)-Net in Base 64 — Upper bound on s

(53−9, 53, 6291450)-Net over F₆₄ — Constructive and digital

(53−9, 53, large)-Net over F₆₄ — Digital