Lee Weight for (u,u + v)-construction of codes over Z4

For a linear code $C$ of length $n$ over $Z_4$‎, ‎The Lee support weight of $C$‎, ‎denoted by $wt_L(C)$‎, ‎is the sum of Lee weights of all columns of $A(C)$, $A(C)$ is $|C| times n$ array of all codewords in $C$‎. ‎## For $1 leq r leq rank(C)$‎, ‎the $r$-th generalized Lee weight with respect to rank (GLWR) for $C$‎, ‎denoted by $d_r^L(C)$‎, ‎is defined as ‎begin{equation*} ‎‎‎‎‎d_r^L(C)=minlbrace wt_L(D); D text{ is a } Z_4-text{submodule of C}, rank(D)=rrbrace‎‎.‎‎‎ ‎end{equation*}‎‎‎‎‎‎ ‎‎ ‎Let $C_i, i=1,2$ be codes over $Z_4$ ‎and ‎‎$‎‎C$ ‎denote‎ ‎‎$(u, u+v)‎$‎‎‎-construction of them. ##In this paper, we obtained $d_1^L(C)$ in terms of $d_1^L(C_1),d_1^L(C_2)$ ‎and ‎we‎ generally obtained an upper bound for $d_r^L(C)$ for all $r‎$‎‎, ‎$1 leq r leq rank(C)$‎.## we found a relationship between ‎$‎‎wt_Lx‎‎$‎, ‎$wt_Ly‎$ ‎and ‎$wt_L(x+y)‎‎$‎‎‎‎‎, for any ‎$‎‎x, y ‎‎in ‎Z_4^n $and we showed that Lee support weight is invariant un‎‎der multiplication by 3