MinT - Best Known (77−13, 77, s)-Nets in Base 64

Best Known (77−13, 77, s)-Nets in Base 64

(77−13, 77, 2798249)-Net over F₆₄ — Constructive and digital

Digital (64, 77, 2798249)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 7, 2049)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁷, 2049, F₆₄, 4, 4) (dual of [(2049, 4), 8189, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(64⁷, 2049, F₆₄, 3, 4) (dual of [(2049, 3), 6140, 5]-NRT-code), using
      - OA 2-folding and stacking [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 (15, 21, 1398100)-net over F₆₄, using
  - s-reduction based on digital (15, 21, 2796201)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64²¹, 2796201, F₆₄, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(64²¹, large, F₆₄, 6) (dual of [large, large−21, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. digital (36, 49, 1398100)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁹, 1398100, F₆₄, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(64⁴⁹, 8388601, F₆₄, 13) (dual of [8388601, 8388552, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(77−13, 77, large)-Net over F₆₄ — Digital

Digital (64, 77, large)-net over F₆₄, using

t-expansion [i] based on digital (63, 77, large)-net over F₆₄, using
- 7 times m-reduction [i] based on digital (63, 84, large)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁴, large, F₆₄, 21) (dual of [large, large−84, 22]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(64⁸¹, large, F₆₄, 21) (dual of [large, large−81, 22]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(77−13, 77, large)-Net in Base 64 — Upper bound on s

There is no (64, 77, large)-net in base 64, because

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

Best Known (77−13, 77, s)-Nets in Base 64

(77−13, 77, 2798249)-Net over F64 — Constructive and digital

(77−13, 77, large)-Net over F64 — Digital

(77−13, 77, large)-Net in Base 64 — Upper bound on s

(77−13, 77, 2798249)-Net over F₆₄ — Constructive and digital

(77−13, 77, large)-Net over F₆₄ — Digital