MinT - Best Known (77−13, 77, s)-Nets in Base 16

Best Known (77−13, 77, s)-Nets in Base 16

(77−13, 77, 2796200)-Net over F₁₆ — Constructive and digital

Digital (64, 77, 2796200)-net over F₁₆, using

16³ times duplication [i] based on digital (61, 74, 2796200)-net over F₁₆, using
- net defined by OOA [i] based on linear OOA(16⁷⁴, 2796200, F₁₆, 14, 13) (dual of [(2796200, 14), 39146726, 14]-NRT-code), using
  - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁷⁴, 8388601, F₁₆, 2, 13) (dual of [(8388601, 2), 16777128, 14]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(16⁷⁴, 8388602, F₁₆, 2, 13) (dual of [(8388602, 2), 16777130, 14]-NRT-code), using
      - trace code [i] based on linear OOA(256³⁷, 4194301, F₂₅₆, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁷, 8388602, F₂₅₆, 13) (dual of [8388602, 8388565, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁷, large, F₂₅₆, 13) (dual of [large, large−37, 14]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(77−13, 77, large)-Net over F₁₆ — Digital

Digital (64, 77, large)-net over F₁₆, using

16⁴ times duplication [i] based on digital (60, 73, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁷³, large, F₁₆, 13) (dual of [large, large−73, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 16¹²âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(77−13, 77, large)-Net in Base 16 — Upper bound on s

There is no (64, 77, large)-net in base 16, because

11 times m-reduction [i] would yield (64, 66, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]