Conflating genes and DNA is your first mistake. Also, DNA is not the hardware, that would be the proteins responsible for executing on the software that is DNA.
First and foremost "genes" do not exist. Genes are a fuzzy concept that people pass around and when talking about DNA. Because of the fuzziness, everyone can claim that genes do what they think they do since they are not constrained by a definition.
DNA is a physical thing, whereas genes are underdefined concepts on top of DNA. Every single thing people call a gene has a different name as well. What someone calls gene might be a protein, might be a transcript, might be the superset of all coding exons of a family of transcripts and so on ... so what is a gene then? I have come to believe that "gene" is a word everyone likes to us yet very few if any understand.
Find any definition of gene and it is either quite circuitous or wrong. For example take the Wikipedia page for gene. The definition devolves into that of a primary transcript. In reality transcripts code for molecules, and multiple transcripts may be labeled as the same gene. Note how genes do not actually exist in the same way as transcript do... the wikipedia page never makes it clear that a gene is a mere label and grouping of the existing transcripts.
Fundamentally, of course, the hardware/software are a metaphor - it does not really work as a hardware nor software.
The point I was making is that the information processing that takes place on DNA is what differentiates the different life forms, not the actual base identity.
Genes certainly do exist. Your cells are not capable of conceptualizing ideas. They need to know where to begin and end transcription and translation to produce specific protein in response to stimuli. Therefore, genes exist as discrete, physiological things.
