MinT - Best Known (82, 105, s)-Nets in Base 16

Best Known (82, 105, s)-Nets in Base 16

(82, 105, 11960)-Net over F₁₆ — Constructive and digital

Digital (82, 105, 11960)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 15, 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 (67, 90, 11915)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁹⁰, 11915, F₁₆, 23, 23) (dual of [(11915, 23), 273955, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(16⁹⁰, 131066, F₁₆, 23) (dual of [131066, 130976, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁹⁰, 131074, F₁₆, 23) (dual of [131074, 130984, 24]-code), using
        trace code [i] based on linear OA(256⁴⁵, 65537, F₂₅₆, 23) (dual of [65537, 65492, 24]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]

(82, 105, 23832)-Net in Base 16 — Constructive

(82, 105, 23832)-net in base 16, using

base change [i] based on digital (47, 70, 23832)-net over F₆₄, using
- 64¹ times duplication [i] based on digital (46, 69, 23832)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁶⁹, 23832, F₆₄, 23, 23) (dual of [(23832, 23), 548067, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(64⁶⁹, 262153, F₆₄, 23) (dual of [262153, 262084, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁶⁹, 262155, F₆₄, 23) (dual of [262155, 262086, 24]-code), using
        constructionÂ X applied to C_e(22) âŠ‚ C_e(19) [i] based on
        linear OA(64⁶⁷, 262144, F₆₄, 23) (dual of [262144, 262077, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(64⁵⁸, 262144, F₆₄, 20) (dual of [262144, 262086, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(64², 11, F₆₄, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(82, 105, 337064)-Net over F₁₆ — Digital

Digital (82, 105, 337064)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 105, large)-Net in Base 16 — Upper bound on s

There is no (82, 105, large)-net in base 16, because

21 times m-reduction [i] would yield (82, 84, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (82, 105, s)-Nets in Base 16

(82, 105, 11960)-Net over F16 — Constructive and digital

(82, 105, 23832)-Net in Base 16 — Constructive

(82, 105, 337064)-Net over F16 — Digital

(82, 105, large)-Net in Base 16 — Upper bound on s

(82, 105, 11960)-Net over F₁₆ — Constructive and digital

(82, 105, 337064)-Net over F₁₆ — Digital