MinT - Best Known (69, 69+10, s)-Nets in Base 16

Best Known (69, 69+10, s)-Nets in Base 16

(69, 69+10, 4399936)-Net over F₁₆ — Constructive and digital

Digital (69, 79, 4399936)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (18, 23, 1044496)-net over F₁₆, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 65281)-net over F₁₆, using
      - s-reduction based on digital (0, 0, s)-net over F₁₆ with arbitrarily large s, using
        arbitrary set of 16⁰ points [i]
    2. digital (0, 0, 65281)-net over F₁₆ (see above)
    3. digital (0, 0, 65281)-net over F₁₆ (see above)
    4. digital (0, 0, 65281)-net over F₁₆ (see above)
    5. digital (0, 0, 65281)-net over F₁₆ (see above)
    6. digital (0, 0, 65281)-net over F₁₆ (see above)
    7. digital (0, 0, 65281)-net over F₁₆ (see above)
    8. digital (0, 0, 65281)-net over F₁₆ (see above)
    9. digital (0, 0, 65281)-net over F₁₆ (see above)
    10. digital (0, 0, 65281)-net over F₁₆ (see above)
    11. digital (0, 0, 65281)-net over F₁₆ (see above)
    12. digital (0, 1, 65281)-net over F₁₆, using
      - s-reduction based on digital (0, 1, s)-net over F₁₆ with arbitrarily large s, using
        construction for strength kÂ =Â 1 [i]
    13. digital (0, 1, 65281)-net over F₁₆ (see above)
    14. digital (0, 1, 65281)-net over F₁₆ (see above)
    15. digital (3, 5, 65281)-net over F₁₆, using
      - s-reduction based on digital (3, 5, 69905)-net over F₁₆, using
        kÂ =Â 2 construction [i]
    16. digital (10, 15, 65281)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16¹⁵, 65281, F₁₆, 5, 5) (dual of [(65281, 5), 326390, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(16¹⁵, 130563, F₁₆, 5) (dual of [130563, 130548, 6]-code), using
        1 times code embedding in larger space [i] based on linear OA(16¹⁴, 130562, F₁₆, 5) (dual of [130562, 130548, 6]-code), using
        trace code [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (46, 56, 3355440)-net over F₁₆, using
  - trace code for nets [i] based on digital (18, 28, 1677720)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256²⁸, 1677720, F₂₅₆, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(256²⁸, 8388600, F₂₅₆, 10) (dual of [8388600, 8388572, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(256²⁸, large, F₂₅₆, 10) (dual of [large, large−28, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(69, 69+10, 4404017)-Net in Base 16 — Constructive

(69, 79, 4404017)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (18, 23, 1048577)-net in base 16, using
  - net defined by OOA [i] based on OOA(16²³, 1048577, S₁₆, 5, 5), using
    - OOA 2-folding and stacking with additional row [i] based on OA(16²³, 2097155, S₁₆, 5), using
      - discarding parts of the base [i] based on linear OA(128¹³, 2097155, F₁₂₈, 5) (dual of [2097155, 2097142, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(128¹³, 2097152, F₁₂₈, 5) (dual of [2097152, 2097139, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(128¹⁰, 2097152, F₁₂₈, 4) (dual of [2097152, 2097142, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (46, 56, 3355440)-net over F₁₆, using
  - trace code for nets [i] based on digital (18, 28, 1677720)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256²⁸, 1677720, F₂₅₆, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(256²⁸, 8388600, F₂₅₆, 10) (dual of [8388600, 8388572, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(256²⁸, large, F₂₅₆, 10) (dual of [large, large−28, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(69, 69+10, large)-Net over F₁₆ — Digital

Digital (69, 79, large)-net over F₁₆, using

t-expansion [i] based on digital (65, 79, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷⁹, large, F₁₆, 14) (dual of [large, large−79, 15]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(69, 69+10, large)-Net in Base 16 — Upper bound on s

There is no (69, 79, large)-net in base 16, because

8 times m-reduction [i] would yield (69, 71, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (69, 69+10, s)-Nets in Base 16

(69, 69+10, 4399936)-Net over F16 — Constructive and digital

(69, 69+10, 4404017)-Net in Base 16 — Constructive

(69, 69+10, large)-Net over F16 — Digital

(69, 69+10, large)-Net in Base 16 — Upper bound on s

(69, 69+10, 4399936)-Net over F₁₆ — Constructive and digital

(69, 69+10, large)-Net over F₁₆ — Digital