MinT - Best Known (125−50, 125, s)-Nets in Base 16

Best Known (125−50, 125, s)-Nets in Base 16

Digital (75, 125, 538)-net over F₁₆, using

1 times m-reduction [i] based on digital (75, 126, 538)-net over F₁₆, using
- trace code for nets [i] based on digital (12, 63, 269)-net over F₂₅₆, using
  - net from sequence [i] based on digital (12, 268)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (75, 125, 1528)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²⁵, 1528, F₁₆, 50) (dual of [1528, 1403, 51]-code), using
- 1402 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 27 times 0, 1, 27 times 0, 1, 30 times 0, 1, 31 times 0, 1, 34 times 0, 1, 35 times 0, 1, 38 times 0, 1, 40 times 0, 1, 42 times 0, 1, 45 times 0, 1, 48 times 0, 1, 50 times 0, 1, 54 times 0, 1, 57 times 0, 1, 60 times 0, 1, 64 times 0, 1, 68 times 0, 1, 72 times 0, 1, 76 times 0, 1, 81 times 0) [i] based on linear OA(16⁵⁰, 51, F₁₆, 50) (dual of [51, 1, 51]-code or 51-arc in PG(49,16)), using
  - dual of repetition code with length 51 [i]

There is no (75, 125, 711421)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 273492 772267 769499 441726 366187 705251 844007 775525 123345 775385 862899 379919 593018 512961 345118 910426 433605 388366 485657 827338 600458 833717 027311 797529 262876 > 16¹²⁵ [i]