MinT - Best Known (7, 7+5, s)-Nets in Base 64

Best Known (7, 7+5, s)-Nets in Base 64

(7, 7+5, 8065)-Net over F₆₄ — Constructive and digital

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
      1. 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]
      2. linear OA(64¹, 4033, F₆₄, 1) (dual of [4033, 4032, 2]-code) (see above)
      3. 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]
      4. linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]

(7, 7+5, 16132)-Net over F₆₄ — Digital

Digital (7, 12, 16132)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹², 16132, F₆₄, 5) (dual of [16132, 16120, 6]-code), using
- generalized (u,Â u+v)-construction [i] based on
  1. 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]
  2. linear OA(64¹, 4033, F₆₄, 1) (dual of [4033, 4032, 2]-code) (see above)
  3. 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]
  4. linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
    - a code by Danev and Olsson [i]

(7, 7+5, 32897)-Net in Base 64 — Constructive

(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
      1. 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]
      2. 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]

(7, 7+5, large)-Net in Base 64 — Upper bound on s

There is no (7, 12, large)-net in base 64, because

3 times m-reduction [i] would yield (7, 9, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]