MinT - Best Known (78, 78+13, s)-Nets in Base 64

Best Known (78, 78+13, s)-Nets in Base 64

(78, 78+13, 5592400)-Net over F₆₄ — Constructive and digital

Digital (78, 91, 5592400)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (57, 70, 2796200)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁷⁰, 2796200, F₆₄, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(64⁷⁰, 8388601, F₆₄, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(64⁷⁰, 8388602, F₆₄, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(64²¹, 4194301, F₆₄, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64²¹, 8388602, F₆₄, 6) (dual of [8388602, 8388581, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(64²¹, large, F₆₄, 6) (dual of [large, large−21, 7]-code) (see above)
        linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(78, 78+13, large)-Net over F₆₄ — Digital

Digital (78, 91, large)-net over F₆₄, using

64³ times duplication [i] based on digital (75, 88, large)-net over F₆₄, using
- t-expansion [i] based on digital (66, 88, large)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁸, large, F₆₄, 22) (dual of [large, large−88, 23]-code), using
    - 3 times code embedding in larger space [i] based on linear OA(64⁸⁵, large, F₆₄, 22) (dual of [large, large−85, 23]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(78, 78+13, large)-Net in Base 64 — Upper bound on s

There is no (78, 91, large)-net in base 64, because

11 times m-reduction [i] would yield (78, 80, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (78, 78+13, s)-Nets in Base 64

(78, 78+13, 5592400)-Net over F64 — Constructive and digital

(78, 78+13, large)-Net over F64 — Digital

(78, 78+13, large)-Net in Base 64 — Upper bound on s

(78, 78+13, 5592400)-Net over F₆₄ — Constructive and digital

(78, 78+13, large)-Net over F₆₄ — Digital