Concatenated codes

This article welcomes suggestions and amendments from others since the author of this article just presents what he thinks is right, and he might be wrong.

Fix n. We will use a binary symmetric channel  n times to transmit n bits.

Firstly, choose a GV code with parameters  over the finite field GF(2)  for the inner code. There are  bits of messages in the GV code.

Secondly, choose a Reed-Solomon code with parameters  over the finite field  for the outer code.

By choosing the above parameters, we can transmit   symbols over the finite field  for the overall code.

The encoding and decoding procedures are done as follows.

The source chooses  symbols over the finite field  to transmit. Encode the  symbols using the  Reed-Solomon code  to obtain a codeword of length  over the finite field . Then, for each symbol in sequence of the Reed-Solomon code, convert it into a -bit sequence and encode the resulting bit sequence using the GV code. Since the length of each bit sequence generated by the GV code is , the total length of the overall sequence is n.

The decoder will first decode each chunk, which is encoded by GV code, to obtain   symbols over the finite field . After that, the decoder will use the Reed-solomon code to try to recover the  symbols.

The choice of d and  will determine the performance of the overall code against errors induced by the channel. More specifically, d and  will determine the minimum Hamming distance of the overall code.