Symmetric product codes

Product codes were introduced by Elias in 1954 and generalized by Tanner in 1981. Recently, a number of generalized product codes have been proposed for forward error-correction in high-speed optical communication. In practice, these codes are decoded by iteratively decoding each of the component codes. Symmetric product codes are a subclass of generalized product codes that use symmetry to reduce the block length of a product code while using the same component code. One example of this subclass, dubbed half-product codes, was introduced by Tanner in 1981 and then generalized by Justesen in 2011. In this paper, we discuss some initial results on symmetric product codes. Our results show that: (i) these codes have a larger normalized minimum distance than the product code from which they are derived, (ii) some small constructions achieve the largest minimum distance possible for a linear code, and (iii) they can have better performance in both the waterfall region and the error floor when compared to a product code of similar length and rate.