MinT - Best Known (6, 11, s)-Nets in Base 64

Best Known (6, 11, s)-Nets in Base 64

(6, 11, 6049)-Net over F₆₄ — Constructive and digital

Digital (6, 11, 6049)-net over F₆₄, using

net defined by OOA [i] based on linear OOA(64¹¹, 6049, F₆₄, 5, 5) (dual of [(6049, 5), 30234, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(64¹¹, 12099, F₆₄, 5) (dual of [12099, 12088, 6]-code), using
  - (u,Â uâˆ’v,Â u+v+w)-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(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]
    3. linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
      - a code by Danev and Olsson [i]

(6, 11, 12099)-Net over F₆₄ — Digital

Digital (6, 11, 12099)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹¹, 12099, F₆₄, 5) (dual of [12099, 12088, 6]-code), using
- (u,Â uâˆ’v,Â u+v+w)-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(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]
  3. linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
    - a code by Danev and Olsson [i]

(6, 11, 32640)-Net in Base 64 — Constructive

(6, 11, 32640)-net in base 64, using

64¹ times duplication [i] based on (5, 10, 32640)-net in base 64, using
- net defined by OOA [i] based on OOA(64¹⁰, 32640, S₆₄, 5, 5), using
  - OOA 2-folding and stacking with additional row [i] based on OA(64¹⁰, 65281, S₆₄, 5), using
    - discarding parts of the base [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
      - a code by Danev and Olsson [i]

(6, 11, large)-Net in Base 64 — Upper bound on s

There is no (6, 11, large)-net in base 64, because

3 times m-reduction [i] would yield (6, 8, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]