2 Overview: Codes From Other Codes

So far in the course, we have seen three types of codes, summarized as follows: 1. Algebraic Codes • Work for worst-case errors. • Have polynomial time encoding and decoding algorithms 2. LDPCs • Work for worst-case errors as well, but a less many errors (smaller distance) compared to the Algebraic Codes. • Have linear time encoding and decoding algorithms and are simpler. 3. Polar Codes • Work for random errors. when viewed in the context of worst-case errors, these codes do not have good distance. • Have O(n log n) encoding and decoding algorithms. Note that this bounds sits somewhere in between the time complexities of the Algebraic Codes and LDPCs. In today’s lecture, we will look into the question of “how can we combine codes to get new codes?”. In this regard, we will consider five types of operations: • Puncturing • Restriction • Tensor • Concatenation