MinT - Best Known (105, 105+16, s)-Nets in Base 16

Best Known (105, 105+16, s)-Nets in Base 16

(105, 105+16, 2113535)-Net over F₁₆ — Constructive and digital

Digital (105, 121, 2113535)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (21, 29, 16385)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16²⁹, 16385, F₁₆, 8, 8) (dual of [(16385, 8), 131051, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(16²⁹, 65540, F₁₆, 8) (dual of [65540, 65511, 9]-code), using
      - constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(16²⁹, 65536, F₁₆, 8) (dual of [65536, 65507, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(16²⁵, 65536, F₁₆, 7) (dual of [65536, 65511, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 16⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(16⁰, 4, F₁₆, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 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 (76, 92, 2097150)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹², 2097150, F₁₆, 18, 16) (dual of [(2097150, 18), 37748608, 17]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OOA(16⁹², 8388601, F₁₆, 2, 16) (dual of [(8388601, 2), 16777110, 17]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(16⁹², 8388602, F₁₆, 2, 16) (dual of [(8388602, 2), 16777112, 17]-NRT-code), using
        trace code [i] based on linear OOA(256⁴⁶, 4194301, F₂₅₆, 2, 16) (dual of [(4194301, 2), 8388556, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256⁴⁶, 8388602, F₂₅₆, 16) (dual of [8388602, 8388556, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁶, large, F₂₅₆, 16) (dual of [large, large−46, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(105, 105+16, large)-Net over F₁₆ — Digital

Digital (105, 121, large)-net over F₁₆, using

t-expansion [i] based on digital (104, 121, large)-net over F₁₆, using
- 6 times m-reduction [i] based on digital (104, 127, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²⁷, large, F₁₆, 23) (dual of [large, large−127, 24]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(105, 105+16, large)-Net in Base 16 — Upper bound on s

There is no (105, 121, large)-net in base 16, because

14 times m-reduction [i] would yield (105, 107, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (105, 105+16, s)-Nets in Base 16

(105, 105+16, 2113535)-Net over F16 — Constructive and digital

(105, 105+16, large)-Net over F16 — Digital

(105, 105+16, large)-Net in Base 16 — Upper bound on s

(105, 105+16, 2113535)-Net over F₁₆ — Constructive and digital

(105, 105+16, large)-Net over F₁₆ — Digital