On the depth spectrum of binary linear codes and their dual

The depth distribution and depth spectrum of a linear code over finite fields were introduced by Etzion. Many interesting properties of the depth of codewords and codes have been obtained. In this paper we study the two extreme cases between the depth spectrum of binary linear codes and their dual codes. Let D(C) be the depth spectrum of a binary linear code C. We obtain a necessary and sufficient condition that D(C)∩D(C⊥)=∅, and a sufficient condition that |D(C)∩D(C⊥)|=⌊n2⌋, where ⌊n2⌋=max{m∈Z:m≤x}.