MinT - Best Known (80−9, 80, s)-Nets in Base 5

Best Known (80−9, 80, s)-Nets in Base 5

(80−9, 80, 2097287)-Net over F₅ — Constructive and digital

Digital (71, 80, 2097287)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 9, 137)-net over F₅, using
  - linear code embedding found in a computer search [i]
2. digital (62, 71, 2097150)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁷¹, 2097150, F₅, 9, 9) (dual of [(2097150, 9), 18874279, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(5⁷¹, 8388601, F₅, 9) (dual of [8388601, 8388530, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁷¹, large, F₅, 9) (dual of [large, large−71, 10]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(80−9, 80, large)-Net over F₅ — Digital

Digital (71, 80, large)-net over F₅, using

1 times m-reduction [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]

(80−9, 80, large)-Net in Base 5 — Upper bound on s

There is no (71, 80, large)-net in base 5, because

7 times m-reduction [i] would yield (71, 73, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]