Erasure and error correcting ability of Parikh matrices

Data transmissions are often subjected to errors due to the noise in the channel. Due to the possibility of errors in the received word, it is important to be able to correct the received word to the word having the best likelihood to be the sent codeword. In this work, we propose the idea of using words sharing the same Parikh matrix (namely, M-equivalent words) as codewords. The Hamming distance for binary and ternary M-equivalent words is studied in detail. Two types of defects in the received word are considered, namely erasure and error. We show that the associated Parikh matrix is able to correct the received word having up to a certain number of erasures and errors. It is also shown that there exists an arbitrarily large M-equivalence class having a minimal Hamming distance of eight, which we reasonably expect to be the largest attainable value.