MinT - Best Known (22−8, 22, s)-Nets in Base 25

Best Known (22−8, 22, s)-Nets in Base 25

(22−8, 22, 3907)-Net over F₂₅ — Constructive and digital

Digital (14, 22, 3907)-net over F₂₅, using

net defined by OOA [i] based on linear OOA(25²², 3907, F₂₅, 8, 8) (dual of [(3907, 8), 31234, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(25²², 15628, F₂₅, 8) (dual of [15628, 15606, 9]-code), using
  - constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
    1. linear OA(25²², 15625, F₂₅, 8) (dual of [15625, 15603, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    2. linear OA(25¹⁹, 15625, F₂₅, 7) (dual of [15625, 15606, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [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]

(22−8, 22, 9743)-Net over F₂₅ — Digital

Digital (14, 22, 9743)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25²², 9743, F₂₅, 8) (dual of [9743, 9721, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(25²², 15625, F₂₅, 8) (dual of [15625, 15603, 9]-code), using
  - an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(22−8, 22, 4503098)-Net in Base 25 — Upper bound on s

There is no (14, 22, 4503099)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 5 684343 943502 941954 917830 102305 > 25²² [i]