MinT - Best Known (66, 66+29, s)-Nets in Base 25

Best Known (66, 66+29, s)-Nets in Base 25

(66, 66+29, 1119)-Net over F₂₅ — Constructive and digital

Digital (66, 95, 1119)-net over F₂₅, using

net defined by OOA [i] based on linear OOA(25⁹⁵, 1119, F₂₅, 29, 29) (dual of [(1119, 29), 32356, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25⁹⁵, 15667, F₂₅, 29) (dual of [15667, 15572, 30]-code), using
  - discarding factors / shortening the dual code based on linear OA(25⁹⁵, 15668, F₂₅, 29) (dual of [15668, 15573, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(17) [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(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]
      3. linear OA(25¹³, 43, F₂₅, 10) (dual of [43, 30, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹³, 52, F₂₅, 10) (dual of [52, 39, 11]-code), using
        extended algebraic-geometric code AG_e(F,41P) [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
        a function field by GarcÃa and Quoos [i]

(66, 66+29, 26064)-Net over F₂₅ — Digital

Digital (66, 95, 26064)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(66, 66+29, large)-Net in Base 25 — Upper bound on s

There is no (66, 95, large)-net in base 25, because

27 times m-reduction [i] would yield (66, 68, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]