MinT - Best Known (77−23, 77, s)-Nets in Base 64

Best Known (77−23, 77, s)-Nets in Base 64

(77−23, 77, 23835)-Net over F₆₄ — Constructive and digital

Digital (54, 77, 23835)-net over F₆₄, using

net defined by OOA [i] based on linear OOA(64⁷⁷, 23835, F₆₄, 23, 23) (dual of [(23835, 23), 548128, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(64⁷⁷, 262186, F₆₄, 23) (dual of [262186, 262109, 24]-code), using
  - discarding factors / shortening the dual code based on linear OA(64⁷⁷, 262187, F₆₄, 23) (dual of [262187, 262110, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(11) [i] based on
      1. linear OA(64⁶⁷, 262144, F₆₄, 23) (dual of [262144, 262077, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(64³⁴, 262144, F₆₄, 12) (dual of [262144, 262110, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      3. linear OA(64¹⁰, 43, F₆₄, 10) (dual of [43, 33, 11]-code or 43-arc in PG(9,64)), using
        discarding factors / shortening the dual code based on linear OA(64¹⁰, 64, F₆₄, 10) (dual of [64, 54, 11]-code or 64-arc in PG(9,64)), using
        Reedâ€“Solomon code RS(54,64) [i]

(77−23, 77, 301410)-Net over F₆₄ — Digital

Digital (54, 77, 301410)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−23, 77, large)-Net in Base 64 — Upper bound on s

There is no (54, 77, large)-net in base 64, because

21 times m-reduction [i] would yield (54, 56, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]