MinT - Best Known (82, 82+11, s)-Nets in Base 5

Best Known (82, 82+11, s)-Nets in Base 5

(82, 82+11, 1677815)-Net over F₅ — Constructive and digital

Digital (82, 93, 1677815)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 12, 95)-net over F₅, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 1, 31)-net over F₅, using
      - s-reduction based on digital (0, 1, s)-net over F₅ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    2. digital (1, 3, 31)-net over F₅, using
      - kÂ =Â 2 construction [i]
    3. digital (3, 8, 33)-net over F₅, using
      - linear code embedding found in a computer search [i]
2. digital (70, 81, 1677720)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁸¹, 1677720, F₅, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(5⁸¹, 8388601, F₅, 11) (dual of [8388601, 8388520, 12]-code), using
      - discarding factors / shortening the dual code 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]

(82, 82+11, large)-Net over F₅ — Digital

Digital (82, 93, large)-net over F₅, using

5³ times duplication [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]

(82, 82+11, large)-Net in Base 5 — Upper bound on s

There is no (82, 93, large)-net in base 5, because

9 times m-reduction [i] would yield (82, 84, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]