MinT - Best Known (82−11, 82, s)-Nets in Base 16

Best Known (82−11, 82, s)-Nets in Base 16

(82−11, 82, 3551282)-Net over F₁₆ — Constructive and digital

Digital (71, 82, 3551282)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (15, 20, 195842)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16²⁰, 195842, F₁₆, 6, 5) (dual of [(195842, 6), 1175032, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(16²⁰, 195843, F₁₆, 2, 5) (dual of [(195843, 2), 391666, 6]-NRT-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(16¹, 65281, F₁₆, 2, 1) (dual of [(65281, 2), 130561, 2]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16¹, s, F₁₆, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(16¹, s, F₁₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OOA(16⁵, 65281, F₁₆, 2, 2) (dual of [(65281, 2), 130557, 3]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁵, 69905, F₁₆, 2, 2) (dual of [(69905, 2), 139805, 3]-NRT-code), using
        appending kth column [i] based on linear OA(16⁵, 69905, F₁₆, 2) (dual of [69905, 69900, 3]-code), using
        Hamming code H(5,16) [i]
        linear OOA(16¹⁴, 65281, F₁₆, 2, 5) (dual of [(65281, 2), 130548, 6]-NRT-code), using
        OOA 2-folding [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 (51, 62, 3355440)-net over F₁₆, using
  - trace code for nets [i] based on digital (20, 31, 1677720)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256³¹, 1677720, F₂₅₆, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(256³¹, 8388601, F₂₅₆, 11) (dual of [8388601, 8388570, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(256³¹, large, F₂₅₆, 11) (dual of [large, large−31, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(82−11, 82, large)-Net over F₁₆ — Digital

Digital (71, 82, large)-net over F₁₆, using

t-expansion [i] based on digital (70, 82, large)-net over F₁₆, using
- 3 times m-reduction [i] based on digital (70, 85, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁵, large, F₁₆, 15) (dual of [large, large−85, 16]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 16¹²âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(82−11, 82, large)-Net in Base 16 — Upper bound on s

There is no (71, 82, large)-net in base 16, because

9 times m-reduction [i] would yield (71, 73, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (82−11, 82, s)-Nets in Base 16

(82−11, 82, 3551282)-Net over F16 — Constructive and digital

(82−11, 82, large)-Net over F16 — Digital

(82−11, 82, large)-Net in Base 16 — Upper bound on s

(82−11, 82, 3551282)-Net over F₁₆ — Constructive and digital

(82−11, 82, large)-Net over F₁₆ — Digital