Ball codes - two-dimensional tail-biting convolutional codes

In this paper we investigate a new class of codes, the 2-D tail-biting convolutional codes (2-D TBCCs). The class of two-dimensional convolutional codes (2-D CCs) is a little-researched subject in coding theory, and tail-biting versions were hardly mentioned, unless they can be represented as a product of two 1-D codes. These codes have interesting geometry since they are the 2-D analog of the 1-D TBCC which their graph is a ring. The result being a code invariant to shifts in 2-D space. We apply algebraic methods in order to find bijective encoders, create parity check matrices and inverse encoders. Next, we discuss minimum distance and weight distribution properties of these codes. We observe that some of these codes exhibit very good codes performance. We then present several novel iterative suboptimal algorithms for soft decoding 2-D CCs, which are based on belief propagation and generalized belief propagation. The results show that the suboptimal algorithms achieve respectable results, in some cases coming as close as 0.4dB from optimal (maximum-likelihood) decoding.