Constructing critical indecomposable codes

Critical indecomposable codes were introduced by Assmus [1], who also gave a recursive construction for these objects. One of the key ingredients in the construction is an auxiliary code, which is an indecomposable code of minimum distance at least 3. In terms of actually being able to construct all critical indecomposable codes, however, Assmus leaves many unanswered questions about these auxiliary codes. In this paper, we provide answers to these questions, including a description of when two equivalent auxiliary codes can yield inequivalent critical indecomposable codes, and results on both the minimum length and the maximum number of critical columns of an auxiliary code. We end with an enumeration of all critical indecomposable codes of dimension at most 10.