MinT - Best Known (77, 77+8, s)-Nets in Base 5

Best Known (77, 77+8, s)-Nets in Base 5

(77, 77+8, 4194900)-Net over F₅ — Constructive and digital

Digital (77, 85, 4194900)-net over F₅, using

5¹ times duplication [i] based on digital (76, 84, 4194900)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (8, 12, 600)-net over F₅, using
    - net defined by OOA [i] based on linear OOA(5¹², 600, F₅, 4, 4) (dual of [(600, 4), 2388, 5]-NRT-code), using
      - appending kth column [i] based on linear OOA(5¹², 600, F₅, 3, 4) (dual of [(600, 3), 1788, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(5¹², 1200, F₅, 4) (dual of [1200, 1188, 5]-code), using
        trace code [i] based on linear OA(25⁶, 600, F₂₅, 4) (dual of [600, 594, 5]-code), using
        1 times truncation [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]
  2. digital (64, 72, 4194300)-net over F₅, using
    - net defined by OOA [i] based on linear OOA(5⁷², 4194300, F₅, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OOA(5⁷², 8388601, F₅, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(5⁷², 8388602, F₅, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
        trace code [i] based on linear OOA(25³⁶, 4194301, F₂₅, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25³⁶, 8388602, F₂₅, 8) (dual of [8388602, 8388566, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(25³⁶, large, F₂₅, 8) (dual of [large, large−36, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(77, 77+8, large)-Net over F₅ — Digital

Digital (77, 85, large)-net over F₅, using

5⁴ times duplication [i] based on digital (73, 81, large)-net over F₅, using
- t-expansion [i] based on digital (71, 81, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁸¹, large, F₅, 10) (dual of [large, large−81, 11]-code), using
    - strength reduction [i] based on linear OA(5⁸¹, large, F₅, 11) (dual of [large, large−81, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 5²⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(77, 77+8, large)-Net in Base 5 — Upper bound on s

There is no (77, 85, large)-net in base 5, because

6 times m-reduction [i] would yield (77, 79, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (77, 77+8, s)-Nets in Base 5

(77, 77+8, 4194900)-Net over F5 — Constructive and digital

(77, 77+8, large)-Net over F5 — Digital

(77, 77+8, large)-Net in Base 5 — Upper bound on s

(77, 77+8, 4194900)-Net over F₅ — Constructive and digital

(77, 77+8, large)-Net over F₅ — Digital