MinT - Best Known (83, 83+10, s)-Nets in Base 5

Best Known (83, 83+10, s)-Nets in Base 5

(83, 83+10, 3355440)-Net over F₅ — Constructive and digital

Digital (83, 93, 3355440)-net over F₅, using

5¹ times duplication [i] based on digital (82, 92, 3355440)-net over F₅, using
- trace code for nets [i] based on digital (36, 46, 1677720)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁴⁶, 1677720, F₂₅, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(25⁴⁶, 8388600, F₂₅, 10) (dual of [8388600, 8388554, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁴⁶, large, F₂₅, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(83, 83+10, large)-Net over F₅ — Digital

Digital (83, 93, large)-net over F₅, using

5³ times duplication [i] based on digital (80, 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]

(83, 83+10, large)-Net in Base 5 — Upper bound on s

There is no (83, 93, large)-net in base 5, because

8 times m-reduction [i] would yield (83, 85, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]