MinT - Best Known (45, 60, s)-Nets in Base 5

Best Known (45, 60, s)-Nets in Base 5

Digital (45, 60, 446)-net over F₅, using

net defined by OOA [i] based on linear OOA(5⁶⁰, 446, F₅, 15, 15) (dual of [(446, 15), 6630, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5⁶⁰, 3123, F₅, 15) (dual of [3123, 3063, 16]-code), using
  - discarding factors / shortening the dual code based on linear OA(5⁶⁰, 3124, F₅, 15) (dual of [3124, 3064, 16]-code), using
    - 1 times truncation [i] based on linear OA(5⁶¹, 3125, F₅, 16) (dual of [3125, 3064, 17]-code), using
      - an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

Digital (45, 60, 2098)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁶⁰, 2098, F₅, 15) (dual of [2098, 2038, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5⁶⁰, 3124, F₅, 15) (dual of [3124, 3064, 16]-code), using
  - 1 times truncation [i] based on linear OA(5⁶¹, 3125, F₅, 16) (dual of [3125, 3064, 17]-code), using
    - an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

There is no (45, 60, 657922)-net in base 5, because

1 times m-reduction [i] would yield (45, 59, 657922)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 173473 129348 940654 041108 371771 308418 450985 > 5⁵⁹ [i]