MinT - Best Known (71−9, 71, s)-Nets in Base 25

Best Known (71−9, 71, s)-Nets in Base 25

(71−9, 71, 6784686)-Net over F₂₅ — Constructive and digital

Digital (62, 71, 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, 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]
18. digital (0, 1, 195314)-net over F₂₅ (see above)
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 (4, 7, 195314)-net over F₂₅, using
  - s-reduction based on digital (4, 7, 391875)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁷, 391875, F₂₅, 3, 3) (dual of [(391875, 3), 1175618, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(25⁷, 391875, F₂₅, 2, 3) (dual of [(391875, 2), 783743, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(25⁷, 391875, F₂₅, 3) (dual of [391875, 391868, 4]-code or 391875-cap in PG(6,25)), using
        product of two caps with tangent hyperplane [i]
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 (32, 41, 2097150)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁴¹, 2097150, F₂₅, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(25⁴¹, 8388601, F₂₅, 9) (dual of [8388601, 8388560, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁴¹, large, F₂₅, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(71−9, 71, large)-Net over F₂₅ — Digital

Digital (62, 71, large)-net over F₂₅, using

t-expansion [i] based on digital (61, 71, large)-net over F₂₅, using
- 6 times m-reduction [i] based on digital (61, 77, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁷⁷, large, F₂₅, 16) (dual of [large, large−77, 17]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25⁷⁶, large, F₂₅, 16) (dual of [large, large−76, 17]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(71−9, 71, large)-Net in Base 25 — Upper bound on s

There is no (62, 71, large)-net in base 25, because

7 times m-reduction [i] would yield (62, 64, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (71−9, 71, s)-Nets in Base 25

(71−9, 71, 6784686)-Net over F25 — Constructive and digital

(71−9, 71, large)-Net over F25 — Digital

(71−9, 71, large)-Net in Base 25 — Upper bound on s

(71−9, 71, 6784686)-Net over F₂₅ — Constructive and digital

(71−9, 71, large)-Net over F₂₅ — Digital