MinT - Best Known (82−15, 82, s)-Nets in Base 16

Best Known (82−15, 82, s)-Nets in Base 16

(82−15, 82, 149842)-Net over F₁₆ — Constructive and digital

Digital (67, 82, 149842)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 11, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (56, 71, 149797)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁷¹, 149797, F₁₆, 15, 15) (dual of [(149797, 15), 2246884, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(16⁷¹, 1048580, F₁₆, 15) (dual of [1048580, 1048509, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁷¹, 1048581, F₁₆, 15) (dual of [1048581, 1048510, 16]-code), using
        constructionÂ X applied to C_e(14) âŠ‚ C_e(13) [i] based on
        linear OA(16⁷¹, 1048576, F₁₆, 15) (dual of [1048576, 1048505, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(16⁶⁶, 1048576, F₁₆, 14) (dual of [1048576, 1048510, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 16⁵âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        linear OA(16⁰, 5, F₁₆, 0) (dual of [5, 5, 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]

(82−15, 82, 299595)-Net in Base 16 — Constructive

(67, 82, 299595)-net in base 16, using

16¹ times duplication [i] based on (66, 81, 299595)-net in base 16, using
- net defined by OOA [i] based on OOA(16⁸¹, 299595, S₁₆, 15, 15), using
  - OOA 7-folding and stacking with additional row [i] based on OA(16⁸¹, 2097166, S₁₆, 15), using
    - discarding factors based on OA(16⁸¹, 2097168, S₁₆, 15), using
      - discarding parts of the base [i] based on linear OA(128⁴⁶, 2097168, F₁₂₈, 15) (dual of [2097168, 2097122, 16]-code), using
        constructionÂ X applied to C([0,7]) âŠ‚ C([0,5]) [i] based on
        linear OA(128⁴³, 2097153, F₁₂₈, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(128³¹, 2097153, F₁₂₈, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(128³, 15, F₁₂₈, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
        discarding factors / shortening the dual code based on linear OA(128³, 128, F₁₂₈, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
        Reedâ€“Solomon code RS(125,128) [i]

(82−15, 82, 4550613)-Net over F₁₆ — Digital

Digital (67, 82, 4550613)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−15, 82, large)-Net in Base 16 — Upper bound on s

There is no (67, 82, large)-net in base 16, because

13 times m-reduction [i] would yield (67, 69, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (82−15, 82, s)-Nets in Base 16

(82−15, 82, 149842)-Net over F16 — Constructive and digital

(82−15, 82, 299595)-Net in Base 16 — Constructive

(82−15, 82, 4550613)-Net over F16 — Digital

(82−15, 82, large)-Net in Base 16 — Upper bound on s

(82−15, 82, 149842)-Net over F₁₆ — Constructive and digital

(82−15, 82, 4550613)-Net over F₁₆ — Digital