MinT - Best Known (29, 29+8, s)-Nets in Base 64

Best Known (29, 29+8, s)-Nets in Base 64

(29, 29+8, 2101310)-Net over F₆₄ — Constructive and digital

Digital (29, 37, 2101310)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 8, 4160)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁸, 4160, F₆₄, 4, 4) (dual of [(4160, 4), 16632, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(64⁸, 4160, F₆₄, 3, 4) (dual of [(4160, 3), 12472, 5]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(64⁰, s, F₆₄, 3, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64⁰, 65, F₆₄, 3, 0) (dual of [(65, 3), 195, 1]-NRT-code) (see above)
        linear OOA(64¹, 65, F₆₄, 3, 1) (dual of [(65, 3), 194, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(64¹, s, F₆₄, 3, 1) with arbitrarily large s, using
        appending 2 arbitrary columns [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, 65, F₆₄, 3, 1) (dual of [(65, 3), 194, 2]-NRT-code) (see above)
        linear OOA(64², 65, F₆₄, 3, 2) (dual of [(65, 3), 193, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;193,64) [i]
        linear OOA(64⁴, 65, F₆₄, 3, 4) (dual of [(65, 3), 191, 5]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(3;191,64) [i]
2. digital (21, 29, 2097150)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64²⁹, 2097150, F₆₄, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(64²⁹, 8388600, F₆₄, 8) (dual of [8388600, 8388571, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(64²⁹, large, F₆₄, 8) (dual of [large, large−29, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(29, 29+8, 2129790)-Net in Base 64 — Constructive

(29, 37, 2129790)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (4, 8, 32640)-net in base 64, using
  - base change [i] based on digital (2, 6, 32640)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁶, 32640, F₂₅₆, 4, 4) (dual of [(32640, 4), 130554, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(256⁶, 65280, F₂₅₆, 4) (dual of [65280, 65274, 5]-code), using
        1 times truncation [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 (21, 29, 2097150)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64²⁹, 2097150, F₆₄, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(64²⁹, 8388600, F₆₄, 8) (dual of [8388600, 8388571, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(64²⁹, large, F₆₄, 8) (dual of [large, large−29, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(29, 29+8, large)-Net over F₆₄ — Digital

Digital (29, 37, large)-net over F₆₄, using

t-expansion [i] based on digital (28, 37, large)-net over F₆₄, using
- 1 times m-reduction [i] based on digital (28, 38, large)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³⁸, large, F₆₄, 10) (dual of [large, large−38, 11]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(64³⁷, large, F₆₄, 10) (dual of [large, large−37, 11]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(29, 29+8, large)-Net in Base 64 — Upper bound on s

There is no (29, 37, large)-net in base 64, because

6 times m-reduction [i] would yield (29, 31, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (29, 29+8, s)-Nets in Base 64

(29, 29+8, 2101310)-Net over F64 — Constructive and digital

(29, 29+8, 2129790)-Net in Base 64 — Constructive

(29, 29+8, large)-Net over F64 — Digital

(29, 29+8, large)-Net in Base 64 — Upper bound on s

(29, 29+8, 2101310)-Net over F₆₄ — Constructive and digital

(29, 29+8, large)-Net over F₆₄ — Digital