MinT - Best Known (83−21, 83, s)-Nets in Base 16

Best Known (83−21, 83, s)-Nets in Base 16

(83−21, 83, 13107)-Net over F₁₆ — Constructive and digital

Digital (62, 83, 13107)-net over F₁₆, using

16¹ times duplication [i] based on digital (61, 82, 13107)-net over F₁₆, using
- net defined by OOA [i] based on linear OOA(16⁸², 13107, F₁₆, 21, 21) (dual of [(13107, 21), 275165, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(16⁸², 131071, F₁₆, 21) (dual of [131071, 130989, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(16⁸², 131074, F₁₆, 21) (dual of [131074, 130992, 22]-code), using
      - trace code [i] based on linear OA(256⁴¹, 65537, F₂₅₆, 21) (dual of [65537, 65496, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(83−21, 83, 83138)-Net over F₁₆ — Digital

Digital (62, 83, 83138)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸³, 83138, F₁₆, 21) (dual of [83138, 83055, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(16⁸³, 131077, F₁₆, 21) (dual of [131077, 130994, 22]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(16⁸², 131076, F₁₆, 21) (dual of [131076, 130994, 22]-code), using
    - trace code [i] based on linear OA(256⁴¹, 65538, F₂₅₆, 21) (dual of [65538, 65497, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [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(256³⁹, 65536, F₂₅₆, 20) (dual of [65536, 65497, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(83−21, 83, large)-Net in Base 16 — Upper bound on s

There is no (62, 83, large)-net in base 16, because

19 times m-reduction [i] would yield (62, 64, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]