MinT - Best Known (78, 78+7, s)-Nets in Base 64

Best Known (78, 78+7, s)-Nets in Base 64

(78, 78+7, large)-Net over F₆₄ — Constructive and digital

Digital (78, 85, large)-net over F₆₄, using

t-expansion [i] based on digital (76, 85, large)-net over F₆₄, using
- 2 times m-reduction [i] based on digital (76, 87, large)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (12, 17, 4194301)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64¹⁷, 4194301, F₆₄, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(64¹⁷, large, F₆₄, 5) (dual of [large, large−17, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
    2. digital (59, 70, 4194366)-net over F₆₄, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 5, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
        digital (54, 65, 4194301)-net over F₆₄, using
        net defined by OOA [i] based on linear OOA(64⁶⁵, 4194301, F₆₄, 15, 11) (dual of [(4194301, 15), 62914450, 12]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OOA(64⁶⁵, large, F₆₄, 3, 11), using
        generalized (u,Â u+v)-construction [i] based on
        linear OOA(64⁷, 2796201, F₆₄, 3, 3) (dual of [(2796201, 3), 8388596, 4]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(64⁷, 8388602, F₆₄, 3, 3) (dual of [(8388602, 3), 25165799, 4]-NRT-code), using
        OOA stacking with additional row [i] based on linear OA(64⁷, large, F₆₄, 3) (dual of [large, large−7, 4]-code), using
        product of projective cap with 64²-ovoid cap in AG(3,Â 64) [i] based on linear OA(64⁴, 4097, F₆₄, 3) (dual of [4097, 4093, 4]-code or 4097-cap in PG(3,64)), using
        ovoid in PG(3,Â 64) [i]
        linear OOA(64¹⁷, 2796201, F₆₄, 3, 5) (dual of [(2796201, 3), 8388586, 6]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(64¹⁷, 4194301, F₆₄, 3, 5) (dual of [(4194301, 3), 12582886, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(64¹⁷, large, F₆₄, 5) (dual of [large, large−17, 6]-code) (see above)
        linear OOA(64⁴¹, 2796201, F₆₄, 3, 11) (dual of [(2796201, 3), 8388562, 12]-NRT-code), using
        OOA 3-folding [i] based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(78, 78+7, large)-Net in Base 64 — Upper bound on s

There is no (78, 85, large)-net in base 64, because

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

Best Known (78, 78+7, s)-Nets in Base 64

(78, 78+7, large)-Net over F64 — Constructive and digital

(78, 78+7, large)-Net in Base 64 — Upper bound on s

(78, 78+7, large)-Net over F₆₄ — Constructive and digital