In general the hamming weight of codewords of error correcting codes is well understood. If I were to write down the $k \times n$ generator matrix, with the span of the rows corresponding to the codewords, for many linear codes, the hamming weight is "somewhat" well understood. I was wondering if I were to look at the span of the columns of the same generator matrix, is there anything that can be said such as it is binomially distributed or etc.? In particular I am interested in BCH codes, I want to understand how the hamming weight of the span of the columns behave (hamming weight of codewords (span of rows) have been studied before by many people). Is there any literature that studies the column span?
Thanks