MinT - Best Known (4, 67, s)-Nets in Base 16

Best Known (4, 67, s)-Nets in Base 16

(4, 67, 45)-Net over F₁₆ — Constructive and digital

Digital (4, 67, 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]

(4, 67, 107)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (4, 67, 108)-net over F₁₆, because

5 times m-reduction [i] would yield digital (4, 62, 108)-net over F₁₆, but
- extracting embedded orthogonal array [i] would yield linear OA(16⁶², 108, F₁₆, 58) (dual of [108, 46, 59]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(16⁶¹, 65, S₁₆, 58), but
      - the linear programming bound shows that MÂ â‰¥ 159214 122701 309768 707410 104386 945873 298246 228915 255775 554254 178010 880553 254912 / 5487 > 16⁶¹ [i]
    2. linear OA(16⁴⁶, 108, F₁₆, 43) (dual of [108, 62, 44]-code), but
      - discarding factors / shortening the dual code would yield linear OA(16⁴⁶, 97, F₁₆, 43) (dual of [97, 51, 44]-code), but
        constructionÂ Y1 [i] would yield
        OA(16⁴⁵, 49, S₁₆, 43), but
        the linear programming bound shows that MÂ â‰¥ 79 689768 125026 220634 634045 411816 077548 174434 353547 313152 / 47 > 16⁴⁵ [i]
        linear OA(16⁵¹, 97, F₁₆, 48) (dual of [97, 46, 49]-code), but
        discarding factors / shortening the dual code would yield linear OA(16⁵¹, 68, F₁₆, 48) (dual of [68, 17, 49]-code), but
        residual code [i] would yield OA(16³, 19, S₁₆, 3), but
        1 times truncation [i] would yield OA(16², 18, S₁₆, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 271 > 16² [i]

(4, 67, 134)-Net in Base 16 — Upper bound on s

There is no (4, 67, 135)-net in base 16, because

3 times m-reduction [i] would yield (4, 64, 135)-net in base 16, but
- extracting embedded orthogonal array [i] would yield OA(16⁶⁴, 135, S₁₆, 60), but
  - the linear programming bound shows that MÂ â‰¥ 927307 381743 870318 843181 626529 192926 580561 440393 777192 457465 581435 826204 073270 718677 661117 803026 211590 373376 / 7 996408 842245 667928 521681 068641 > 16⁶⁴ [i]