error correcting huffman code I am looking for a code that can correct errors with variable and limited length like huffman code. I am not an expert in coding theory. Is there any code or related literature on this?
 A: Variable Length Error Correction (VLEC) codes do exist, and they are used in some applications where source coding/channel coding separation may not be the best way to go and both compression and error correction is required. Here are a number of papers along this direction:
Buttigieg and Farrell, On variable-length error-correcting codes
Proceedings of 1994 IEEE International Symposium on Information Theory
Savari and Kliewer, When Huffman meets Hamming:  A Class of Optimal
Variable-Length Error Correcting Codes, Data Compression Conference 2010,
available here
Wu, Chen, Alajaji and Han, On the Design of
Variable-Length Error-Correcting Codes, *IEEE Trans. Communications, 61(9),2013.
The quote below is from the last paper:

Also shown
in this paper is that our VLEC system outperforms traditional
separate source/channel coding systems of similar overall rate
at low to medium SNRs with the benefit of considerably
smaller decoding complexity.

The codes are found by search, but this is not a problem due to the existence of efficient search algorithms, and other code families are also the result of search algorithms. The decoding algorithms tend to be quite fast.
One must not forget that a lot of the mathematically beautiful algebraic codes usually do not fare so well in applications, since most channels of interest (wireless mobile) these days are not i.i.d. and gaussian but time varying.
