MinT - Best Known (122−29, 122, s)-Nets in Base 4

Best Known (122−29, 122, s)-Nets in Base 4

Digital (93, 122, 1032)-net over F₄, using

4² times duplication [i] based on digital (91, 120, 1032)-net over F₄, using
- trace code for nets [i] based on digital (1, 30, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (93, 122, 1596)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²², 1596, F₄, 29) (dual of [1596, 1474, 30]-code), using
- 1473 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (8, 4, 2, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 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, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 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, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 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, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 32 times 0, 1, 35 times 0, 1, 36 times 0, 1, 38 times 0, 1, 40 times 0, 1, 42 times 0, 1, 45 times 0, 1, 47 times 0, 1, 49 times 0, 1, 52 times 0, 1, 55 times 0, 1, 58 times 0, 1, 61 times 0, 1, 63 times 0, 1, 67 times 0, 1, 71 times 0, 1, 75 times 0) [i] based on linear OA(4²⁹, 30, F₄, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,4)), using
  - dual of repetition code with length 30 [i]

There is no (93, 122, 321989)-net in base 4, because

1 times m-reduction [i] would yield (93, 121, 321989)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 067642 310114 575155 131661 206844 453446 425549 073498 841873 237723 122739 942432 > 4¹²¹ [i]