MinT - Best Known (100−31, 100, s)-Nets in Base 25

Best Known (100−31, 100, s)-Nets in Base 25

(100−31, 100, 1044)-Net over F₂₅ — Constructive and digital

Digital (69, 100, 1044)-net over F₂₅, using

net defined by OOA [i] based on linear OOA(25¹⁰⁰, 1044, F₂₅, 31, 31) (dual of [(1044, 31), 32264, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(25¹⁰⁰, 15661, F₂₅, 31) (dual of [15661, 15561, 32]-code), using
  - discarding factors / shortening the dual code based on linear OA(25¹⁰⁰, 15664, F₂₅, 31) (dual of [15664, 15564, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(20) [i] based on
      1. linear OA(25⁸⁸, 15625, F₂₅, 31) (dual of [15625, 15537, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. 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]
      3. linear OA(25¹², 39, F₂₅, 9) (dual of [39, 27, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹², 52, F₂₅, 9) (dual of [52, 40, 10]-code), using
        extended algebraic-geometric code AG_e(F,42P) [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
        a function field by GarcÃa and Quoos [i]

(100−31, 100, 22944)-Net over F₂₅ — Digital

Digital (69, 100, 22944)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(100−31, 100, large)-Net in Base 25 — Upper bound on s

There is no (69, 100, large)-net in base 25, because

29 times m-reduction [i] would yield (69, 71, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]