MinT - Best Known (34−6, 34, s)-Nets in Base 49

Best Known (34−6, 34, s)-Nets in Base 49

(34−6, 34, 7686501)-Net over F₄₉ — Constructive and digital

Digital (28, 34, 7686501)-net over F₄₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 120100)-net over F₄₉, using
  - s-reduction based on digital (0, 0, s)-net over F₄₉ with arbitrarily large s, using
    - arbitrary set of 49⁰ points [i]
2. digital (0, 0, 120100)-net over F₄₉ (see above)
3. digital (0, 0, 120100)-net over F₄₉ (see above)
4. digital (0, 0, 120100)-net over F₄₉ (see above)
5. digital (0, 0, 120100)-net over F₄₉ (see above)
6. digital (0, 0, 120100)-net over F₄₉ (see above)
7. digital (0, 0, 120100)-net over F₄₉ (see above)
8. digital (0, 0, 120100)-net over F₄₉ (see above)
9. digital (0, 0, 120100)-net over F₄₉ (see above)
10. digital (0, 0, 120100)-net over F₄₉ (see above)
11. digital (0, 0, 120100)-net over F₄₉ (see above)
12. digital (0, 0, 120100)-net over F₄₉ (see above)
13. digital (0, 0, 120100)-net over F₄₉ (see above)
14. digital (0, 0, 120100)-net over F₄₉ (see above)
15. digital (0, 0, 120100)-net over F₄₉ (see above)
16. digital (0, 0, 120100)-net over F₄₉ (see above)
17. digital (0, 0, 120100)-net over F₄₉ (see above)
18. digital (0, 0, 120100)-net over F₄₉ (see above)
19. digital (0, 0, 120100)-net over F₄₉ (see above)
20. digital (0, 0, 120100)-net over F₄₉ (see above)
21. digital (0, 0, 120100)-net over F₄₉ (see above)
22. digital (0, 0, 120100)-net over F₄₉ (see above)
23. digital (0, 0, 120100)-net over F₄₉ (see above)
24. digital (0, 0, 120100)-net over F₄₉ (see above)
25. digital (0, 0, 120100)-net over F₄₉ (see above)
26. digital (0, 0, 120100)-net over F₄₉ (see above)
27. digital (0, 0, 120100)-net over F₄₉ (see above)
28. digital (0, 0, 120100)-net over F₄₉ (see above)
29. digital (0, 0, 120100)-net over F₄₉ (see above)
30. digital (0, 0, 120100)-net over F₄₉ (see above)
31. digital (0, 0, 120100)-net over F₄₉ (see above)
32. digital (0, 0, 120100)-net over F₄₉ (see above)
33. digital (0, 0, 120100)-net over F₄₉ (see above)
34. digital (0, 0, 120100)-net over F₄₉ (see above)
35. digital (0, 0, 120100)-net over F₄₉ (see above)
36. digital (0, 0, 120100)-net over F₄₉ (see above)
37. digital (0, 0, 120100)-net over F₄₉ (see above)
38. digital (0, 0, 120100)-net over F₄₉ (see above)
39. digital (0, 0, 120100)-net over F₄₉ (see above)
40. digital (0, 0, 120100)-net over F₄₉ (see above)
41. digital (0, 0, 120100)-net over F₄₉ (see above)
42. digital (0, 0, 120100)-net over F₄₉ (see above)
43. digital (0, 0, 120100)-net over F₄₉ (see above)
44. digital (0, 1, 120100)-net over F₄₉, using
  - s-reduction based on digital (0, 1, s)-net over F₄₉ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
45. digital (0, 1, 120100)-net over F₄₉ (see above)
46. digital (0, 1, 120100)-net over F₄₉ (see above)
47. digital (2, 4, 120100)-net over F₄₉, using
  - kÂ =Â 2 construction [i]
48. digital (3, 6, 120200)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49⁶, 120200, F₄₉, 3, 3) (dual of [(120200, 3), 360594, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(49⁶, 120200, F₄₉, 2, 3) (dual of [(120200, 2), 240394, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(49⁶, 120200, F₄₉, 3) (dual of [120200, 120194, 4]-code or 120200-cap in PG(5,49)), using
        large caps in PG(5,Â b) [i]
49. digital (15, 21, 1921601)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49²¹, 1921601, F₄₉, 6, 6) (dual of [(1921601, 6), 11529585, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(49²¹, 5764803, F₄₉, 6) (dual of [5764803, 5764782, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(49²¹, 5764805, F₄₉, 6) (dual of [5764805, 5764784, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(49²¹, 5764801, F₄₉, 6) (dual of [5764801, 5764780, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(49¹⁷, 5764801, F₄₉, 5) (dual of [5764801, 5764784, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(49⁰, 4, F₄₉, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(49⁰, s, F₄₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(34−6, 34, large)-Net over F₄₉ — Digital

Digital (28, 34, large)-net over F₄₉, using

t-expansion [i] based on digital (26, 34, large)-net over F₄₉, using
- Gilbertâ€“VarÅ¡amov bound [i]

(34−6, 34, large)-Net in Base 49 — Upper bound on s

There is no (28, 34, large)-net in base 49, because

4 times m-reduction [i] would yield (28, 30, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]

Best Known (34−6, 34, s)-Nets in Base 49

(34−6, 34, 7686501)-Net over F49 — Constructive and digital

(34−6, 34, large)-Net over F49 — Digital

(34−6, 34, large)-Net in Base 49 — Upper bound on s

(34−6, 34, 7686501)-Net over F₄₉ — Constructive and digital

(34−6, 34, large)-Net over F₄₉ — Digital