MinT - Best Known (67−15, 67, s)-Nets in Base 16

Best Known (67−15, 67, s)-Nets in Base 16

(67−15, 67, 18758)-Net over F₁₆ — Constructive and digital

Digital (52, 67, 18758)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 9, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (43, 58, 18725)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁵⁸, 18725, F₁₆, 15, 15) (dual of [(18725, 15), 280817, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(16⁵⁸, 131076, F₁₆, 15) (dual of [131076, 131018, 16]-code), using
      - trace code [i] based on linear OA(256²⁹, 65538, F₂₅₆, 15) (dual of [65538, 65509, 16]-code), using
        constructionÂ X applied to C_e(14) âŠ‚ C_e(13) [i] based on
        linear OA(256²⁹, 65536, F₂₅₆, 15) (dual of [65536, 65507, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(256²⁷, 65536, F₂₅₆, 14) (dual of [65536, 65509, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [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]

(67−15, 67, 37450)-Net in Base 16 — Constructive

(52, 67, 37450)-net in base 16, using

16¹ times duplication [i] based on (51, 66, 37450)-net in base 16, using
- base change [i] based on digital (29, 44, 37450)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁴⁴, 37450, F₆₄, 15, 15) (dual of [(37450, 15), 561706, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(64⁴⁴, 262151, F₆₄, 15) (dual of [262151, 262107, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁴⁴, 262152, F₆₄, 15) (dual of [262152, 262108, 16]-code), using
        constructionÂ X applied to C([0,7]) âŠ‚ C([0,6]) [i] based on
        linear OA(64⁴³, 262145, F₆₄, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(64³⁷, 262145, F₆₄, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(64¹, 7, F₆₄, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(67−15, 67, 233321)-Net over F₁₆ — Digital

Digital (52, 67, 233321)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−15, 67, large)-Net in Base 16 — Upper bound on s

There is no (52, 67, large)-net in base 16, because

13 times m-reduction [i] would yield (52, 54, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (67−15, 67, s)-Nets in Base 16

(67−15, 67, 18758)-Net over F16 — Constructive and digital

(67−15, 67, 37450)-Net in Base 16 — Constructive

(67−15, 67, 233321)-Net over F16 — Digital

(67−15, 67, large)-Net in Base 16 — Upper bound on s

(67−15, 67, 18758)-Net over F₁₆ — Constructive and digital

(67−15, 67, 233321)-Net over F₁₆ — Digital