The depth distribution-a new characterization for linear codes

We apply the well known operator of sequences, the derivative D, on codewords of linear codes. The depth of a codeword c is the smallest integer i such that Dⁱc (the derivative applied i consecutive times) is zero. We show that the depth distribution of the nonzero codewords of an [n,k] linear code consists of exactly k nonzero values, and its generator matrix can be constructed from any k nonzero codewords with distinct depths. Interesting properties of some linear codes, and a way to partition equivalent codes into new equivalence classes are also discussed