MinT - Best Known (89−21, 89, s)-Nets in Base 16

Best Known (89−21, 89, s)-Nets in Base 16

(89−21, 89, 13109)-Net over F₁₆ — Constructive and digital

Digital (68, 89, 13109)-net over F₁₆, using

16¹ times duplication [i] based on digital (67, 88, 13109)-net over F₁₆, using
- net defined by OOA [i] based on linear OOA(16⁸⁸, 13109, F₁₆, 21, 21) (dual of [(13109, 21), 275201, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(16⁸⁸, 131091, F₁₆, 21) (dual of [131091, 131003, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(16⁸⁸, 131096, F₁₆, 21) (dual of [131096, 131008, 22]-code), using
      - trace code [i] based on linear OA(256⁴⁴, 65548, F₂₅₆, 21) (dual of [65548, 65504, 22]-code), using
        constructionÂ X applied to C([0,10]) âŠ‚ C([0,8]) [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]
        linear OA(256³³, 65537, F₂₅₆, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(256³, 11, F₂₅₆, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
        discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]

(89−21, 89, 131098)-Net over F₁₆ — Digital

Digital (68, 89, 131098)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁸⁹, 131098, F₁₆, 21) (dual of [131098, 131009, 22]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(16⁸⁸, 131096, F₁₆, 21) (dual of [131096, 131008, 22]-code), using
    - trace code [i] based on linear OA(256⁴⁴, 65548, F₂₅₆, 21) (dual of [65548, 65504, 22]-code), using
      - constructionÂ X applied to C([0,10]) âŠ‚ C([0,8]) [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]
        linear OA(256³³, 65537, F₂₅₆, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(256³, 11, F₂₅₆, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
        discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]
  2. linear OA(16⁸⁸, 131097, F₁₆, 20) (dual of [131097, 131009, 21]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 16⁸⁸Â > V_b^sâˆ’1(kâˆ’1)Â = 3 120874 868726 688853 779430 071290 678489 632759 740757 943884 008505 056295 331521 773524 653399 553785 716334 462441 [i]
  3. linear OA(16⁰, 1, F₁₆, 0) (dual of [1, 1, 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]

(89−21, 89, large)-Net in Base 16 — Upper bound on s

There is no (68, 89, large)-net in base 16, because

19 times m-reduction [i] would yield (68, 70, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (89−21, 89, s)-Nets in Base 16

(89−21, 89, 13109)-Net over F16 — Constructive and digital

(89−21, 89, 131098)-Net over F16 — Digital

(89−21, 89, large)-Net in Base 16 — Upper bound on s

(89−21, 89, 13109)-Net over F₁₆ — Constructive and digital

(89−21, 89, 131098)-Net over F₁₆ — Digital