MinT - Best Known (59, 80, s)-Nets in Base 25

Best Known (59, 80, s)-Nets in Base 25

(59, 80, 1687)-Net over F₂₅ — Constructive and digital

Digital (59, 80, 1687)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (9, 19, 125)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25¹⁹, 125, F₂₅, 10, 10) (dual of [(125, 10), 1231, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(25¹⁹, 625, F₂₅, 10) (dual of [625, 606, 11]-code), using
      - an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
2. digital (40, 61, 1562)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶¹, 1562, F₂₅, 21, 21) (dual of [(1562, 21), 32741, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(25⁶¹, 15621, F₂₅, 21) (dual of [15621, 15560, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁶¹, 15625, F₂₅, 21) (dual of [15625, 15564, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(59, 80, 135172)-Net over F₂₅ — Digital

Digital (59, 80, 135172)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 80, large)-Net in Base 25 — Upper bound on s

There is no (59, 80, large)-net in base 25, because

19 times m-reduction [i] would yield (59, 61, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]