MinT - Best Known (4, 8, s)-Nets in Base 49

Best Known (4, 8, s)-Nets in Base 49

(4, 8, 2450)-Net over F₄₉ — Constructive and digital

Digital (4, 8, 2450)-net over F₄₉, using

net defined by OOA [i] based on linear OOA(49⁸, 2450, F₄₉, 4, 4) (dual of [(2450, 4), 9792, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(49⁸, 2450, F₄₉, 3, 4) (dual of [(2450, 3), 7342, 5]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(49⁰, s, F₄₉, 3, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
    2. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    3. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    4. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    5. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    6. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    7. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    8. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    9. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    10. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    11. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    12. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    13. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    14. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    15. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    16. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    17. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    18. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    19. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    20. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    21. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    22. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    23. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    24. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    25. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    26. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    27. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    28. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    29. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    30. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    31. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    32. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    33. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    34. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    35. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    36. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    37. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    38. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    39. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    40. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    41. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    42. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    43. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    44. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    45. linear OOA(49⁰, 50, F₄₉, 3, 0) (dual of [(50, 3), 150, 1]-NRT-code) (see above)
    46. linear OOA(49¹, 50, F₄₉, 3, 1) (dual of [(50, 3), 149, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(49¹, s, F₄₉, 3, 1) with arbitrarily large s, using
        appending 2 arbitrary columns [i] based on linear OA(49¹, s, F₄₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    47. linear OOA(49¹, 50, F₄₉, 3, 1) (dual of [(50, 3), 149, 2]-NRT-code) (see above)
    48. linear OOA(49², 50, F₄₉, 3, 2) (dual of [(50, 3), 148, 3]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(3;148,49) [i]
    49. linear OOA(49⁴, 50, F₄₉, 3, 4) (dual of [(50, 3), 146, 5]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(3;146,49) [i]

(4, 8, 2501)-Net over F₄₉ — Digital

Digital (4, 8, 2501)-net over F₄₉, using

net defined by OOA [i] based on linear OOA(49⁸, 2501, F₄₉, 4, 4) (dual of [(2501, 4), 9996, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(49⁸, 2501, F₄₉, 3, 4) (dual of [(2501, 3), 7495, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49⁸, 2501, F₄₉, 4) (dual of [2501, 2493, 5]-code), using
    - 97 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 96 times 0) [i] based on linear OA(49⁷, 2403, F₄₉, 4) (dual of [2403, 2396, 5]-code), using
      - constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(49⁷, 2401, F₄₉, 4) (dual of [2401, 2394, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(49⁵, 2401, F₄₉, 3) (dual of [2401, 2396, 4]-code or 2401-cap in PG(4,49)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 2400Â = 49²âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(49⁰, 2, F₄₉, 0) (dual of [2, 2, 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]

(4, 8, 169846)-Net in Base 49 — Upper bound on s

There is no (4, 8, 169847)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 33 233111 896225 > 49⁸ [i]

Best Known (4, 8, s)-Nets in Base 49

(4, 8, 2450)-Net over F49 — Constructive and digital

(4, 8, 2501)-Net over F49 — Digital

(4, 8, 169846)-Net in Base 49 — Upper bound on s

(4, 8, 2450)-Net over F₄₉ — Constructive and digital

(4, 8, 2501)-Net over F₄₉ — Digital