MinT - Best Known (70, 70+21, s)-Nets in Base 16

Best Known (70, 70+21, s)-Nets in Base 16

(70, 70+21, 13110)-Net over F₁₆ — Constructive and digital

Digital (70, 91, 13110)-net over F₁₆, using

net defined by OOA [i] based on linear OOA(16⁹¹, 13110, F₁₆, 21, 21) (dual of [(13110, 21), 275219, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(16⁹¹, 131101, F₁₆, 21) (dual of [131101, 131010, 22]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(16⁹⁰, 131100, F₁₆, 21) (dual of [131100, 131010, 22]-code), using
    - trace code [i] based on linear OA(256⁴⁵, 65550, F₂₅₆, 21) (dual of [65550, 65505, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(15) [i] based on
        linear OA(256⁴¹, 65536, F₂₅₆, 21) (dual of [65536, 65495, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(256³¹, 65536, F₂₅₆, 16) (dual of [65536, 65505, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(256⁴, 14, F₂₅₆, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
        discarding factors / shortening the dual code based on linear OA(256⁴, 256, F₂₅₆, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
        Reedâ€“Solomon code RS(252,256) [i]

(70, 70+21, 166720)-Net over F₁₆ — Digital

Digital (70, 91, 166720)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+21, large)-Net in Base 16 — Upper bound on s

There is no (70, 91, large)-net in base 16, because

19 times m-reduction [i] would yield (70, 72, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]