MinT - Best Known (26−5, 26, s)-Nets in Base 25

Best Known (26−5, 26, s)-Nets in Base 25

(26−5, 26, 4882949)-Net over F₂₅ — Constructive and digital

Digital (21, 26, 4882949)-net over F₂₅, using

net defined by OOA [i] based on linear OOA(25²⁶, 4882949, F₂₅, 6, 5) (dual of [(4882949, 6), 29297668, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(25²⁶, 4882950, F₂₅, 2, 5) (dual of [(4882950, 2), 9765874, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(25⁰, s, F₂₅, 2, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
    2. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    3. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    4. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    5. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    6. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    7. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    8. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    9. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    10. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    11. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    12. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    13. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    14. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    15. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    16. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    17. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    18. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    19. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    20. linear OOA(25⁰, 195318, F₂₅, 2, 0) (dual of [(195318, 2), 390636, 1]-NRT-code) (see above)
    21. linear OOA(25¹, 195318, F₂₅, 2, 1) (dual of [(195318, 2), 390635, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(25¹, s, F₂₅, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(25¹, s, F₂₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    22. linear OOA(25¹, 195318, F₂₅, 2, 1) (dual of [(195318, 2), 390635, 2]-NRT-code) (see above)
    23. linear OOA(25¹, 195318, F₂₅, 2, 1) (dual of [(195318, 2), 390635, 2]-NRT-code) (see above)
    24. linear OOA(25⁵, 195318, F₂₅, 2, 2) (dual of [(195318, 2), 390631, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(25⁵, 406901, F₂₅, 2, 2) (dual of [(406901, 2), 813797, 3]-NRT-code), using
        appending kth column [i] based on linear OA(25⁵, 406901, F₂₅, 2) (dual of [406901, 406896, 3]-code), using
        Hamming code H(5,25) [i]
    25. linear OOA(25¹⁸, 195318, F₂₅, 2, 5) (dual of [(195318, 2), 390618, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(25¹⁸, 390636, F₂₅, 5) (dual of [390636, 390618, 6]-code), using
        constructionÂ X4 applied to C([0,2]) âŠ‚ C([0,1]) [i] based on
        linear OA(25¹⁷, 390626, F₂₅, 5) (dual of [390626, 390609, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626Â | 25⁸âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(25⁹, 390626, F₂₅, 3) (dual of [390626, 390617, 4]-code or 390626-cap in PG(8,25)), using the expurgated narrow-sense BCH-code C(I) with length 390626Â | 25⁸âˆ’1, defining interval IÂ =Â [0,1], and minimum distance dÂ â‰¥ |{−1,0,1}|+1Â =Â 4 (BCH-bound) [i]
        linear OA(25⁹, 10, F₂₅, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,25)), using
        dual of repetition code with length 10 [i]
        linear OA(25¹, 10, F₂₅, 1) (dual of [10, 9, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹, 25, F₂₅, 1) (dual of [25, 24, 2]-code), using
        Reedâ€“Solomon code RS(24,25) [i]

(26−5, 26, large)-Net over F₂₅ — Digital

Digital (21, 26, large)-net over F₂₅, using

t-expansion [i] based on digital (20, 26, large)-net over F₂₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25²⁶, large, F₂₅, 6) (dual of [large, large−26, 7]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(26−5, 26, large)-Net in Base 25 — Upper bound on s

There is no (21, 26, large)-net in base 25, because

3 times m-reduction [i] would yield (21, 23, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (26−5, 26, s)-Nets in Base 25

(26−5, 26, 4882949)-Net over F25 — Constructive and digital

(26−5, 26, large)-Net over F25 — Digital

(26−5, 26, large)-Net in Base 25 — Upper bound on s

(26−5, 26, 4882949)-Net over F₂₅ — Constructive and digital

(26−5, 26, large)-Net over F₂₅ — Digital