MinT - Best Known (117−27, 117, s)-Nets in Base 4

Best Known (117−27, 117, s)-Nets in Base 4

Digital (90, 117, 1036)-net over F₄, using

4¹ times duplication [i] based on digital (89, 116, 1036)-net over F₄, using
- trace code for nets [i] based on digital (2, 29, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (90, 117, 1814)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁷, 1814, F₄, 27) (dual of [1814, 1697, 28]-code), using
- 1696 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (7, 4, 2, 2, 1, 2, 1, 1, 1, 0, 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, 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, 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, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 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, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 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, 47 times 0, 1, 50 times 0, 1, 53 times 0, 1, 56 times 0, 1, 59 times 0, 1, 63 times 0, 1, 66 times 0, 1, 69 times 0, 1, 74 times 0, 1, 77 times 0, 1, 82 times 0, 1, 87 times 0, 1, 92 times 0) [i] based on linear OA(4²⁷, 28, F₄, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,4)), using
  - dual of repetition code with length 28 [i]

There is no (90, 117, 445146)-net in base 4, because

1 times m-reduction [i] would yield (90, 116, 445146)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6901 750080 711946 696170 100760 072893 951034 715174 076320 809498 000529 027844 > 4¹¹⁶ [i]