system
Given the following post, respond as if you're commenting on Reddit:
human
Does our body store every antibody to every infection/disease we've ever had?

If so, where are they stored and how is it "coded"?
gpt
The answer is a yes/no.

First, not all disease is infectious, nor produces an adaptive immune response; while some disease results from an adaptive immune response.  Furthermore not all adaptive response results in antibody body production as the predominant response.

Assuming we have an immune response we can have one that is predominated by cell mediated immunity, cd-8 t-cells, macrophages, NK cells and the like; or a response dominated by plasma cell proliferation and antibody production. 

In the process of either of these responses memory cells, either T or B are produced.  There are other processes as well in which genes coding for antibody undergo mutation and refinement so as they are more specific, or show a higher affinity to an antigenic molecule on the infectious agent.  

The "code" as you call it, is the rearranged DNA that codes for antibody that that cell and it's progeny will produce.  It survives past infection in the form of memory cells, which have a variable persistence, and in some cases persist for a few months to years, to a decade or longer.  Factors affecting which cells will persist for differing time periods are not well understood.  This is demonstrated in current vaccination practices, some vaccines will impart an immunity that lasts a few years, whereas others produce an immunity lasting decades.  Similarly some immunizations will produce a b-cell response only and others a t-cell, those with a t-cell response produce a stronger immunity

so long answer short, there is a memory; but it often "forgotten."