MinT - Best Known (63, 63+12, s)-Nets in Base 64

Best Known (63, 63+12, s)-Nets in Base 64

(63, 63+12, 2800375)-Net over F₆₄ — Constructive and digital

Digital (63, 75, 2800375)-net over F₆₄, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 9, 4175)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁹, 4175, F₆₄, 4, 4) (dual of [(4175, 4), 16691, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(64⁹, 4175, F₆₄, 3, 4) (dual of [(4175, 3), 12516, 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(64⁵, 80, F₆₄, 3, 4) (dual of [(80, 3), 235, 5]-NRT-code), using
        extended algebraic-geometric NRT-code AG_e(3;F,235P) [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (15, 21, 1398100)-net over F₆₄, using
  - s-reduction based on digital (15, 21, 2796201)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64²¹, 2796201, F₆₄, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(64²¹, large, F₆₄, 6) (dual of [large, large−21, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. digital (33, 45, 1398100)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁵, 1398100, F₆₄, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(64⁴⁵, 8388600, F₆₄, 12) (dual of [8388600, 8388555, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(63, 63+12, 2812520)-Net in Base 64 — Constructive

(63, 75, 2812520)-net in base 64, using

net defined by OOA [i] based on OOA(64⁷⁵, 2812520, S₆₄, 15, 12), using
- OOA 2-folding and stacking with additional row [i] based on OOA(64⁷⁵, 5625041, S₆₄, 3, 12), using
  - discarding factors based on OOA(64⁷⁵, 5625042, S₆₄, 3, 12), using
    - discarding parts of the base [i] based on linear OOA(256⁵⁶, 5625042, F₂₅₆, 3, 12) (dual of [(5625042, 3), 16875070, 13]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(256⁶, 32640, F₂₅₆, 3, 4) (dual of [(32640, 3), 97914, 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]
        linear OOA(256¹⁶, 2796201, F₂₅₆, 3, 6) (dual of [(2796201, 3), 8388587, 7]-NRT-code), using
        OOA 3-folding [i] based on linear OA(256¹⁶, large, F₂₅₆, 6) (dual of [large, large−16, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OOA(256³⁴, 2796201, F₂₅₆, 3, 12) (dual of [(2796201, 3), 8388569, 13]-NRT-code), using
        OOA 3-folding [i] based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(63, 63+12, large)-Net over F₆₄ — Digital

Digital (63, 75, large)-net over F₆₄, using

9 times m-reduction [i] based on digital (63, 84, large)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁴, large, F₆₄, 21) (dual of [large, large−84, 22]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(64⁸¹, large, F₆₄, 21) (dual of [large, large−81, 22]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(63, 63+12, large)-Net in Base 64 — Upper bound on s

There is no (63, 75, large)-net in base 64, because

10 times m-reduction [i] would yield (63, 65, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (63, 63+12, s)-Nets in Base 64

(63, 63+12, 2800375)-Net over F64 — Constructive and digital

(63, 63+12, 2812520)-Net in Base 64 — Constructive

(63, 63+12, large)-Net over F64 — Digital

(63, 63+12, large)-Net in Base 64 — Upper bound on s

(63, 63+12, 2800375)-Net over F₆₄ — Constructive and digital

(63, 63+12, large)-Net over F₆₄ — Digital