MinT - Best Known (18, 21, s)-Nets in Base 64

Best Known (18, 21, s)-Nets in Base 64

(18, 21, large)-Net over F₆₄ — Constructive and digital

Digital (18, 21, large)-net over F₆₄, using

t-expansion [i] based on digital (17, 21, large)-net over F₆₄, using
- 1 times m-reduction [i] based on digital (17, 22, large)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 65)-net over F₆₄, using
      - kÂ =Â 2 construction [i]
    2. digital (15, 20, 8388602)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64²⁰, 8388602, F₆₄, 6, 5) (dual of [(8388602, 6), 50331592, 6]-NRT-code), using
        OOA stacking with additional row [i] based on linear OOA(64²⁰, large, F₆₄, 2, 5), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(64⁰, s, F₆₄, 2, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64⁰, 131073, F₆₄, 2, 0) (dual of [(131073, 2), 262146, 1]-NRT-code) (see above)
        linear OOA(64¹, 131073, F₆₄, 2, 1) (dual of [(131073, 2), 262145, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(64¹, s, F₆₄, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] 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 OOA(64¹, 131073, F₆₄, 2, 1) (dual of [(131073, 2), 262145, 2]-NRT-code) (see above)
        linear OOA(64¹, 131073, F₆₄, 2, 1) (dual of [(131073, 2), 262145, 2]-NRT-code) (see above)
        linear OOA(64⁴, 131073, F₆₄, 2, 2) (dual of [(131073, 2), 262142, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(64⁴, 266305, F₆₄, 2, 2) (dual of [(266305, 2), 532606, 3]-NRT-code), using
        appending kth column [i] based on linear OA(64⁴, 266305, F₆₄, 2) (dual of [266305, 266301, 3]-code), using
        Hamming code H(4,64) [i]
        linear OOA(64¹³, 131073, F₆₄, 2, 5) (dual of [(131073, 2), 262133, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64¹³, 262146, F₆₄, 5) (dual of [262146, 262133, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹³, 262147, F₆₄, 5) (dual of [262147, 262134, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(64¹³, 262144, F₆₄, 5) (dual of [262144, 262131, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(64¹⁰, 262144, F₆₄, 4) (dual of [262144, 262134, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 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]

(18, 21, large)-Net in Base 64 — Upper bound on s

There is no (18, 21, large)-net in base 64, because

1 times m-reduction [i] would yield (18, 20, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (18, 21, s)-Nets in Base 64

(18, 21, large)-Net over F64 — Constructive and digital

(18, 21, large)-Net in Base 64 — Upper bound on s

(18, 21, large)-Net over F₆₄ — Constructive and digital