MinT - Best Known (93−9, 93, s)-Nets in Base 5

Best Known (93−9, 93, s)-Nets in Base 5

(93−9, 93, 4194600)-Net over F₅ — Constructive and digital

Digital (84, 93, 4194600)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 11, 300)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹¹, 300, F₅, 4, 4) (dual of [(300, 4), 1189, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(5¹¹, 300, F₅, 3, 4) (dual of [(300, 3), 889, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(5¹¹, 600, F₅, 4) (dual of [600, 589, 5]-code), using
        base reduction for projective spaces (embedding PG(5,25) in PG(10,5)) [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 (73, 82, 4194300)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁸², 4194300, F₅, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(5⁸², 8388601, F₅, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(5⁸², 8388602, F₅, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
        trace code [i] based on linear OOA(25⁴¹, 4194301, F₂₅, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25⁴¹, 8388602, F₂₅, 9) (dual of [8388602, 8388561, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁴¹, large, F₂₅, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(93−9, 93, large)-Net over F₅ — Digital

Digital (84, 93, large)-net over F₅, using

5³ times duplication [i] based on digital (81, 90, large)-net over F₅, using
- t-expansion [i] based on digital (79, 90, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁰, large, F₅, 11) (dual of [large, large−90, 12]-code), using
    - 9 times code embedding in larger space [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]

(93−9, 93, large)-Net in Base 5 — Upper bound on s

There is no (84, 93, large)-net in base 5, because

7 times m-reduction [i] would yield (84, 86, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (93−9, 93, s)-Nets in Base 5

(93−9, 93, 4194600)-Net over F5 — Constructive and digital

(93−9, 93, large)-Net over F5 — Digital

(93−9, 93, large)-Net in Base 5 — Upper bound on s

(93−9, 93, 4194600)-Net over F₅ — Constructive and digital

(93−9, 93, large)-Net over F₅ — Digital