MinT - Best Known (78, 101, s)-Nets in Base 4

Best Known (78, 101, s)-Nets in Base 4

Digital (78, 101, 1036)-net over F₄, using

4¹ times duplication [i] based on digital (77, 100, 1036)-net over F₄, using
- trace code for nets [i] based on digital (2, 25, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (78, 101, 1764)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰¹, 1764, F₄, 23) (dual of [1764, 1663, 24]-code), using
- 1662 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 33 times 0, 1, 34 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 45 times 0, 1, 49 times 0, 1, 51 times 0, 1, 55 times 0, 1, 59 times 0, 1, 63 times 0, 1, 67 times 0, 1, 71 times 0, 1, 76 times 0, 1, 81 times 0, 1, 87 times 0, 1, 92 times 0, 1, 99 times 0, 1, 105 times 0) [i] based on linear OA(4²³, 24, F₄, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
  - dual of repetition code with length 24 [i]

There is no (78, 101, 486584)-net in base 4, because

1 times m-reduction [i] would yield (78, 100, 486584)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 606938 432145 905033 348831 685148 113028 364764 557882 423944 346870 > 4¹⁰⁰ [i]