MinT - Best Known (82−15, 82, s)-Nets in Base 5

Best Known (82−15, 82, s)-Nets in Base 5

(82−15, 82, 2256)-Net over F₅ — Constructive and digital

Digital (67, 82, 2256)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 10, 25)-net over F₅, using
  - linear code embedding found in a computer search [i]
2. digital (57, 72, 2231)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁷², 2231, F₅, 15, 15) (dual of [(2231, 15), 33393, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(5⁷², 15618, F₅, 15) (dual of [15618, 15546, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁷², 15624, F₅, 15) (dual of [15624, 15552, 16]-code), using
        1 times truncation [i] based on linear OA(5⁷³, 15625, F₅, 16) (dual of [15625, 15552, 17]-code), using
        an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 5⁶âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(82−15, 82, 18772)-Net over F₅ — Digital

Digital (67, 82, 18772)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−15, 82, large)-Net in Base 5 — Upper bound on s

There is no (67, 82, large)-net in base 5, because

13 times m-reduction [i] would yield (67, 69, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]