MinT - Best Known (42, 53, s)-Nets in Base 64

Best Known (42, 53, s)-Nets in Base 64

(42, 53, 1685785)-Net over F₆₄ — Constructive and digital

Digital (42, 53, 1685785)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 12, 8065)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64¹², 8065, F₆₄, 5, 5) (dual of [(8065, 5), 40313, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(64¹², 16131, F₆₄, 5) (dual of [16131, 16119, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(64¹², 16132, F₆₄, 5) (dual of [16132, 16120, 6]-code), using
        generalized (u,Â u+v)-construction [i] based on
        linear OA(64¹, 4033, F₆₄, 1) (dual of [4033, 4032, 2]-code), using
        discarding factors / shortening the dual code 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 OA(64¹, 4033, F₆₄, 1) (dual of [4033, 4032, 2]-code) (see above)
        linear OA(64³, 4033, F₆₄, 2) (dual of [4033, 4030, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(64³, 4161, F₆₄, 2) (dual of [4161, 4158, 3]-code), using
        Hamming code H(3,64) [i]
        linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (30, 41, 1677720)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(42, 53, 1710617)-Net in Base 64 — Constructive

(42, 53, 1710617)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (7, 12, 32897)-net in base 64, using
  - base change [i] based on digital (4, 9, 32897)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁹, 32897, F₂₅₆, 5, 5) (dual of [(32897, 5), 164476, 6]-NRT-code), using
      - appending kth column [i] based on linear OOA(256⁹, 32897, F₂₅₆, 4, 5) (dual of [(32897, 4), 131579, 6]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(256², 257, F₂₅₆, 4, 2) (dual of [(257, 4), 1026, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;1026,256) [i]
        linear OOA(256⁷, 32640, F₂₅₆, 4, 5) (dual of [(32640, 4), 130553, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [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 (30, 41, 1677720)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(42, 53, large)-Net over F₆₄ — Digital

Digital (42, 53, large)-net over F₆₄, using

t-expansion [i] based on digital (41, 53, large)-net over F₆₄, using
- 2 times m-reduction [i] based on digital (41, 55, large)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁵, large, F₆₄, 14) (dual of [large, large−55, 15]-code), using
    - 2 times code embedding in larger space [i] based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(42, 53, large)-Net in Base 64 — Upper bound on s

There is no (42, 53, large)-net in base 64, because

9 times m-reduction [i] would yield (42, 44, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (42, 53, s)-Nets in Base 64

(42, 53, 1685785)-Net over F64 — Constructive and digital

(42, 53, 1710617)-Net in Base 64 — Constructive

(42, 53, large)-Net over F64 — Digital

(42, 53, large)-Net in Base 64 — Upper bound on s

(42, 53, 1685785)-Net over F₆₄ — Constructive and digital

(42, 53, large)-Net over F₆₄ — Digital