MinT - Best Known (70−22, 70, s)-Nets in Base 25

Best Known (70−22, 70, s)-Nets in Base 25

(70−22, 70, 1422)-Net over F₂₅ — Constructive and digital

Digital (48, 70, 1422)-net over F₂₅, using

1 times m-reduction [i] based on digital (48, 71, 1422)-net over F₂₅, using
- net defined by OOA [i] based on linear OOA(25⁷¹, 1422, F₂₅, 23, 23) (dual of [(1422, 23), 32635, 24]-NRT-code), using
  - OOA 11-folding and stacking with additional row [i] based on linear OA(25⁷¹, 15643, F₂₅, 23) (dual of [15643, 15572, 24]-code), using
    - discarding factors / shortening the dual code based on linear OA(25⁷¹, 15644, F₂₅, 23) (dual of [15644, 15573, 24]-code), using
      - constructionÂ X applied to C_e(22) âŠ‚ C_e(17) [i] based on
        linear OA(25⁶⁷, 15625, F₂₅, 23) (dual of [15625, 15558, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(25⁵², 15625, F₂₅, 18) (dual of [15625, 15573, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(25⁴, 19, F₂₅, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,25)), using
        discarding factors / shortening the dual code based on linear OA(25⁴, 25, F₂₅, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
        Reedâ€“Solomon code RS(21,25) [i]

(70−22, 70, 16533)-Net over F₂₅ — Digital

Digital (48, 70, 16533)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(70−22, 70, large)-Net in Base 25 — Upper bound on s

There is no (48, 70, large)-net in base 25, because

20 times m-reduction [i] would yield (48, 50, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]