MinT - Best Known (33−12, 33, s)-Nets in Base 64

Best Known (33−12, 33, s)-Nets in Base 64

(33−12, 33, 1618)-Net over F₆₄ — Constructive and digital

Digital (21, 33, 1618)-net over F₆₄, using

net defined by OOA [i] based on linear OOA(64³³, 1618, F₆₄, 12, 12) (dual of [(1618, 12), 19383, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(64³³, 9708, F₆₄, 12) (dual of [9708, 9675, 13]-code), using
  - discarding factors / shortening the dual code based on linear OA(64³³, 9709, F₆₄, 12) (dual of [9709, 9676, 13]-code), using
    - the BCH-code C(I) with length 9709Â | 64³âˆ’1, defining interval IÂ =Â {1,10,19,…,100}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(33−12, 33, 10923)-Net in Base 64 — Constructive

(21, 33, 10923)-net in base 64, using

64¹ times duplication [i] based on (20, 32, 10923)-net in base 64, using
- base change [i] based on digital (12, 24, 10923)-net over F₂₅₆, using
  - 256¹ times duplication [i] based on digital (11, 23, 10923)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256²³, 10923, F₂₅₆, 12, 12) (dual of [(10923, 12), 131053, 13]-NRT-code), using
      - OA 6-folding and stacking [i] based on linear OA(256²³, 65538, F₂₅₆, 12) (dual of [65538, 65515, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(256²³, 65536, F₂₅₆, 12) (dual of [65536, 65513, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(256²¹, 65536, F₂₅₆, 11) (dual of [65536, 65515, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(33−12, 33, 20433)-Net over F₆₄ — Digital

Digital (21, 33, 20433)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(33−12, 33, 21847)-Net in Base 64

(21, 33, 21847)-net in base 64, using

64¹ times duplication [i] based on (20, 32, 21847)-net in base 64, using
- base change [i] based on digital (12, 24, 21847)-net over F₂₅₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256²⁴, 21847, F₂₅₆, 3, 12) (dual of [(21847, 3), 65517, 13]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(256²⁴, 65541, F₂₅₆, 12) (dual of [65541, 65517, 13]-code), using
      - constructionÂ X applied to C_e(11) âŠ‚ C_e(9) [i] based on
        linear OA(256²³, 65536, F₂₅₆, 12) (dual of [65536, 65513, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(256¹⁹, 65536, F₂₅₆, 10) (dual of [65536, 65517, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(256¹, 5, F₂₅₆, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(256¹, s, F₂₅₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(33−12, 33, large)-Net in Base 64 — Upper bound on s

There is no (21, 33, large)-net in base 64, because

10 times m-reduction [i] would yield (21, 23, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]