MinT - Best Known (83−19, 83, s)-Nets in Base 4

Best Known (83−19, 83, s)-Nets in Base 4

Digital (64, 83, 1032)-net over F₄, using

1 times m-reduction [i] based on digital (64, 84, 1032)-net over F₄, using
- trace code for nets [i] based on digital (1, 21, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

Digital (64, 83, 1513)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸³, 1513, F₄, 19) (dual of [1513, 1430, 20]-code), using
- 1429 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 3, 2, 1, 1, 1, 1, 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, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 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, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 24 times 0, 1, 26 times 0, 1, 28 times 0, 1, 30 times 0, 1, 33 times 0, 1, 36 times 0, 1, 39 times 0, 1, 43 times 0, 1, 46 times 0, 1, 49 times 0, 1, 54 times 0, 1, 58 times 0, 1, 63 times 0, 1, 68 times 0, 1, 74 times 0, 1, 80 times 0, 1, 86 times 0, 1, 94 times 0, 1, 101 times 0, 1, 110 times 0) [i] based on linear OA(4¹⁹, 20, F₄, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,4)), using
  - dual of repetition code with length 20 [i]

There is no (64, 83, 422726)-net in base 4, because

1 times m-reduction [i] would yield (64, 82, 422726)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23 384259 633504 302017 331546 707013 163794 227100 294104 > 4⁸² [i]