A lemma on binomial coefficients and applications to Lee weights modulo 2^e of codes over ${\mathbb{Z}_4}$

Abstract

In this work, we prove that if C is a free ${\mathbb{Z}_4}$-module of rank k in ${\mathbb{Z}_4^n}$, and ${j\in \mathbb{Z}}$ and e ≥ 1, then the number of codewords in C with Lee weight congruent to j modulo 2^e is divisible by ${2^{\left \lfloor \large{\frac{k-2^{e-2}}{2^{e-2}}} \right \rfloor}}$. We prove this result by introducing a lemma and applying the lemma in one of the theorems proved by Wilson. The method used is different than the one used in our previous work on Lee weight enumerators in which more general results were obtained. Moreover, Wilson’s methods are used to prove that the results obtained are sharp by calculating the power of 2 that divides the number of codewords in the trivial code ${\mathbb{Z}_{4}^k}$ with Lee weight congruent to j modulo 2^e.