MinT - Best Known (53, 82, s)-Nets in Base 25

Best Known (53, 82, s)-Nets in Base 25

(53, 82, 1116)-Net over F₂₅ — Constructive and digital

Digital (53, 82, 1116)-net over F₂₅, using

net defined by OOA [i] based on linear OOA(25⁸², 1116, F₂₅, 29, 29) (dual of [(1116, 29), 32282, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25⁸², 15625, F₂₅, 29) (dual of [15625, 15543, 30]-code), using
  - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(53, 82, 7814)-Net over F₂₅ — Digital

Digital (53, 82, 7814)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(25⁸², 7814, F₂₅, 2, 29) (dual of [(7814, 2), 15546, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25⁸², 15628, F₂₅, 29) (dual of [15628, 15546, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
    1. linear OA(25⁸², 15625, F₂₅, 29) (dual of [15625, 15543, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(25⁷⁹, 15625, F₂₅, 28) (dual of [15625, 15546, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(25⁰, 3, F₂₅, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁰, s, F₂₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(53, 82, large)-Net in Base 25 — Upper bound on s

There is no (53, 82, large)-net in base 25, because

27 times m-reduction [i] would yield (53, 55, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]