MinT - Best Known (7, 103, s)-Nets in Base 16

Best Known (7, 103, s)-Nets in Base 16

(7, 103, 65)-Net over F₁₆ — Constructive and digital

Digital (7, 103, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 103, 65)-net over F₁₆, using
- net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(7, 103, 161)-Net over F₁₆ — Upper bound on s (digital)

There is no digital (7, 103, 162)-net over F₁₆, because

extracting embedded orthogonal array [i] would yield linear OA(16¹⁰³, 162, F₁₆, 96) (dual of [162, 59, 97]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(16¹⁰², 107, F₁₆, 96) (dual of [107, 5, 97]-code), but
    - constructionÂ Y1 [i] would yield
      1. OA(16¹⁰¹, 103, S₁₆, 96), but
        the (dual) Plotkin bound shows that MÂ â‰¥ 4627 391781 531740 192663 407156 229397 278798 832780 749968 534992 541566 920840 538654 179420 776324 473078 006993 933138 484175 163508 129792 / 97 > 16¹⁰¹ [i]
      2. OA(16⁵, 107, S₁₆, 4), but
        discarding factors would yield OA(16⁵, 97, S₁₆, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1 049056 > 16⁵ [i]
  2. OA(16⁵⁹, 162, S₁₆, 55), but
    - discarding factors would yield OA(16⁵⁹, 158, S₁₆, 55), but
      - the linear programming bound shows that MÂ â‰¥ 1 751872 128079 872472 547941 838539 530895 802398 111785 402312 961189 355293 686055 441948 303058 733784 687094 938824 867840 / 15 835355 243697 262421 313219 417467 218513 > 16⁵⁹ [i]

(7, 103, 166)-Net in Base 16 — Upper bound on s

There is no (7, 103, 167)-net in base 16, because

extracting embedded orthogonal array [i] would yield OA(16¹⁰³, 167, S₁₆, 96), but
- the linear programming bound shows that MÂ â‰¥ 4936 004781 058610 000402 920240 044284 199962 024946 752402 983805 254322 107257 975940 570413 498163 830539 140460 548898 957165 794340 484190 576976 657040 846834 979703 416183 872239 691259 445248 / 466658 986218 602823 769577 306013 035733 427115 535877 > 16¹⁰³ [i]