Production and performance analysis of geometric construction codes

Recently (Altay G., 2006), a new binary linear block code construction technique, Geometric Construction (GC) codes was proposed that provide the full-information rate for all the Hamming distance-4 even codes. The generator matrix of these codes has very useful properties such that it contains the lowest density of ones, it has quasi-cyclic and regular structure, and it is composed of smaller group generator matrices. A code with cyclic or quasicyclic property can be encoded with less complexity and GC codes have this property. Therefore, it is advantages from practical point of view as it provides flexibility in encoder and decoder implementations. Extended Hamming codes and distance-4 Reed Muller (RM) codes are subsets of our GC construction codes since they generate codes as the power of 2 in length, whereas GC codes generates the codes as the multiple of 2 in length. Since the sizes of GC codes are multiple of 2 for Hamming distance-4, they can also be employed as component codes in BTC, TPC and PA codes. The error performances of GC codes over Additive White Gaussian Noise (AWGN) channel are evaluated. Sum-Product Algorithm (SPA) is an iterative decoding algorithm and it is extremely efficient for decoding LDPC codes. We employed SPA for decoding GC codes as they are very suitable for decoding using the parity control matrix of GC codes. Since GC codes are high rate, the parity control matrix of GC codes has the smallest possible size for Hamming distance-4 codes. Consequently the complexity of decoding process is reduced. The generator matrix of GC codes can easily be converted into systematic form to obtain paritycontrol matrix of it using Gauss-Jordan algorithm (Noble B., 1969). In order to keep the paper self-contained we first give an overview of construction for GC codes and then present the simulation results. Keywords: Linear block codes, error performance, sum-product algorithm.