On the Extended Lee Weights Modulo \(2^e\) of Linear Codes Over \(\mathbb{Z}_{2}\)

Abstract

In this work, linear codes over \(\mathbb{Z}_{2^s}\) are considered together with the extended Lee weight, which is defined as
\[w_L(a) = \begin{cases}
a & \text{if } a \leq 2^{s-1}, \\
2^s – x & \text{if } a > 2^{s-1}.
\end{cases}\]
The ideas used by Wilson and Yildiz are employed to obtain divisibility properties for sums involving binomial coefficients and the extended Lee weight. These results are then used to find bounds on the power of 2 that divides the number of codewords whose Lee weights fall in the same congruence class modulo \(2^e\). Comparisons are made with the results for the trivial code and the results for the homogeneous weight.