MinT - Best Known (52−12, 52, s)-Nets in Base 16

Best Known (52−12, 52, s)-Nets in Base 16

(52−12, 52, 21863)-Net over F₁₆ — Constructive and digital

Digital (40, 52, 21863)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (34, 46, 21846)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁴⁶, 21846, F₁₆, 12, 12) (dual of [(21846, 12), 262106, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(16⁴⁶, 131076, F₁₆, 12) (dual of [131076, 131030, 13]-code), using
      - trace code [i] based on linear OA(256²³, 65538, F₂₅₆, 12) (dual of [65538, 65515, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(256²³, 65536, F₂₅₆, 12) (dual of [65536, 65513, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(256²¹, 65536, F₂₅₆, 11) (dual of [65536, 65515, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(256⁰, 2, F₂₅₆, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁰, s, F₂₅₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(52−12, 52, 43691)-Net in Base 16 — Constructive

(40, 52, 43691)-net in base 16, using

16¹ times duplication [i] based on (39, 51, 43691)-net in base 16, using
- base change [i] based on digital (22, 34, 43691)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³⁴, 43691, F₆₄, 12, 12) (dual of [(43691, 12), 524258, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(64³⁴, 262146, F₆₄, 12) (dual of [262146, 262112, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(64³⁴, 262147, F₆₄, 12) (dual of [262147, 262113, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(64³⁴, 262144, F₆₄, 12) (dual of [262144, 262110, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(64³¹, 262144, F₆₄, 11) (dual of [262144, 262113, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(52−12, 52, 161117)-Net over F₁₆ — Digital

Digital (40, 52, 161117)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(52−12, 52, large)-Net in Base 16 — Upper bound on s

There is no (40, 52, large)-net in base 16, because

10 times m-reduction [i] would yield (40, 42, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (52−12, 52, s)-Nets in Base 16

(52−12, 52, 21863)-Net over F16 — Constructive and digital

(52−12, 52, 43691)-Net in Base 16 — Constructive

(52−12, 52, 161117)-Net over F16 — Digital

(52−12, 52, large)-Net in Base 16 — Upper bound on s

(52−12, 52, 21863)-Net over F₁₆ — Constructive and digital

(52−12, 52, 161117)-Net over F₁₆ — Digital