MinT - Best Known (68, 68+9, s)-Nets in Base 5

Best Known (68, 68+9, s)-Nets in Base 5

(68, 68+9, 2097177)-Net over F₅ — Constructive and digital

Digital (68, 77, 2097177)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 27)-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]

(68, 68+9, large)-Net over F₅ — Digital

Digital (68, 77, large)-net over F₅, using

5⁵ times duplication [i] based on digital (63, 72, large)-net over F₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁷², large, F₅, 9) (dual of [large, large−72, 10]-code), using
  - 1 times code embedding in larger space [i] 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]

(68, 68+9, large)-Net in Base 5 — Upper bound on s

There is no (68, 77, large)-net in base 5, because

7 times m-reduction [i] would yield (68, 70, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]