MinT - Best Known (68−8, 68, s)-Nets in Base 5

Best Known (68−8, 68, s)-Nets in Base 5

(68−8, 68, 2097194)-Net over F₅ — Constructive and digital

Digital (60, 68, 2097194)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 7, 44)-net over F₅, using
  - linear code embedding found in a computer search [i]
2. digital (53, 61, 2097150)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁶¹, 2097150, F₅, 8, 8) (dual of [(2097150, 8), 16777139, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(5⁶¹, 8388600, F₅, 8) (dual of [8388600, 8388539, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁶¹, large, F₅, 8) (dual of [large, large−61, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(68−8, 68, large)-Net over F₅ — Digital

Digital (60, 68, large)-net over F₅, using

5⁶ times duplication [i] based on digital (54, 62, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁶², large, F₅, 8) (dual of [large, large−62, 9]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(5⁶¹, large, F₅, 8) (dual of [large, large−61, 9]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(68−8, 68, large)-Net in Base 5 — Upper bound on s

There is no (60, 68, large)-net in base 5, because

6 times m-reduction [i] would yield (60, 62, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]