MinT - Best Known (63−8, 63, s)-Nets in Base 25

Best Known (63−8, 63, s)-Nets in Base 25

(63−8, 63, 6784686)-Net over F₂₅ — Constructive and digital

Digital (55, 63, 6784686)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 195314)-net over F₂₅, using
  - s-reduction based on digital (0, 0, s)-net over F₂₅ with arbitrarily large s, using
    - arbitrary set of 25⁰ points [i]
2. digital (0, 0, 195314)-net over F₂₅ (see above)
3. digital (0, 0, 195314)-net over F₂₅ (see above)
4. digital (0, 0, 195314)-net over F₂₅ (see above)
5. digital (0, 0, 195314)-net over F₂₅ (see above)
6. digital (0, 0, 195314)-net over F₂₅ (see above)
7. digital (0, 0, 195314)-net over F₂₅ (see above)
8. digital (0, 0, 195314)-net over F₂₅ (see above)
9. digital (0, 0, 195314)-net over F₂₅ (see above)
10. digital (0, 0, 195314)-net over F₂₅ (see above)
11. digital (0, 0, 195314)-net over F₂₅ (see above)
12. digital (0, 0, 195314)-net over F₂₅ (see above)
13. digital (0, 0, 195314)-net over F₂₅ (see above)
14. digital (0, 0, 195314)-net over F₂₅ (see above)
15. digital (0, 0, 195314)-net over F₂₅ (see above)
16. digital (0, 0, 195314)-net over F₂₅ (see above)
17. digital (0, 0, 195314)-net over F₂₅ (see above)
18. digital (0, 1, 195314)-net over F₂₅, using
  - s-reduction based on digital (0, 1, s)-net over F₂₅ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
19. digital (0, 1, 195314)-net over F₂₅ (see above)
20. digital (0, 1, 195314)-net over F₂₅ (see above)
21. digital (0, 1, 195314)-net over F₂₅ (see above)
22. digital (3, 5, 195314)-net over F₂₅, using
  - s-reduction based on digital (3, 5, 406901)-net over F₂₅, using
    - kÂ =Â 2 construction [i]
23. digital (3, 5, 195314)-net over F₂₅ (see above)
24. digital (9, 13, 195314)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25¹³, 195314, F₂₅, 4, 4) (dual of [(195314, 4), 781243, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(25¹³, 390628, F₂₅, 4) (dual of [390628, 390615, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(25¹³, 390629, F₂₅, 4) (dual of [390629, 390616, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(25¹³, 390625, F₂₅, 4) (dual of [390625, 390612, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(25⁹, 390625, F₂₅, 3) (dual of [390625, 390616, 4]-code or 390625-cap in PG(8,25)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(25⁰, 4, F₂₅, 0) (dual of [4, 4, 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]
25. digital (28, 36, 2097150)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25³⁶, 2097150, F₂₅, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(25³⁶, 8388600, F₂₅, 8) (dual of [8388600, 8388564, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(25³⁶, large, F₂₅, 8) (dual of [large, large−36, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(63−8, 63, large)-Net over F₂₅ — Digital

Digital (55, 63, large)-net over F₂₅, using

t-expansion [i] based on digital (53, 63, large)-net over F₂₅, using
- 4 times m-reduction [i] based on digital (53, 67, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁶⁷, large, F₂₅, 14) (dual of [large, large−67, 15]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25⁶⁶, large, F₂₅, 14) (dual of [large, large−66, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(63−8, 63, large)-Net in Base 25 — Upper bound on s

There is no (55, 63, large)-net in base 25, because

6 times m-reduction [i] would yield (55, 57, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (63−8, 63, s)-Nets in Base 25

(63−8, 63, 6784686)-Net over F25 — Constructive and digital

(63−8, 63, large)-Net over F25 — Digital

(63−8, 63, large)-Net in Base 25 — Upper bound on s

(63−8, 63, 6784686)-Net over F₂₅ — Constructive and digital

(63−8, 63, large)-Net over F₂₅ — Digital