In this paper, we consider fast decoding of soft-decision (SD) Hamming codes as inner codes in concatenated forward error-correction (FEC) schemes for high-speed optical communication. The goal is single FPGA implementations at speeds of 400 Gb/s and beyond. A low complexity maximum a posteriori (MAP) probability decoding is applied to a (128,120) Hamming code. Chase decoding of a (128,119) Hamming code is also implemented. The VHDL designs for both decoding schemes are presented. The FEC performance and FPGA resource utilization are investigated and compared. Synthesis results indicate that, both the Chase and the MAP decoder leave sufficient resources available to also accommodate a powerful outer hard decision code, on a single FPGA. Furthermore, MAP decoding of (128,120) Hamming code features lower hardware complexity and provides a higher data throughput.

Decoding of FEC (forward error correction) for optical communication beyond 1 Tb/s is investigated. A configurable single FPGA solution is presented having configurations supporting bit-rates in the range from 40 Gb/s to 1.6 Tb/s. The design allows for trade-offs of bit-rate, footprint, and latency within the resources of the FPGA. A proof-of-concept lab experiment at 40 Gb/s was conducted and pre-FEC — post-FEC performance validated with simulated results.

Modern mobile telecommunication systems, such as 3GPP LTE, make use of Hybrid Automatic Repeat reQuest (HARQ) for efficient and reliable communication between base stationsand mobile terminals. To this purpose, marginal posterior probabilities of the received bits are stored in the form of log-likelihood ratios (LLR) in order to combine information sent across different transmissions due to requests. To mitigate the effects of ever-increasing data rates that call for larger HARQ memory, vector quantization (VQ) is investigated as a technique for temporary compression of LLRs on the terminal. A capacity analysis leads to using maximum mutual information (MMI) as optimality criterion and in turn Kullback-Leibler (KL) divergence as distortion measure. Simulations run based on an LTE-like system have proven that VQ can be implemented in a computationally simple way at low rates of 2-3 bits per LLR value without compromising the system throughput.

In image compression context-based entropy coding is commonly used. A critical issue to the performance of context-based image coding is how to resolve the conflict of a desire for large templates to model high-order statistic dependency of the pixels and the problem of context dilution due to insufficient sample statistics of a given input image. We consider the problem of finding the optimal quantizer Q that quantizes the K-dimensional causal context Ct = (Xt-t1,Xt-t2,...,X t-tK) of a source symbol Xt into one of a set of conditioning states. The optimality of context quantization is defined to be the minimum static or minimum adaptive code length of given a data set. For a binary source alphabet an optimal context quantizer can be computed exactly by a fast dynamic programming algorithm. Faster approximation solutions are also proposed. In case of m-ary source alphabet a random variable can be decomposed into a sequence of binary decisions, each of which is coded using optimal context quantization designed for the corresponding binary random variable. This optimized coding scheme is applied to digital maps and alpha-plane sequences. The proposed optimal context quantization technique can also be used to establish a lower bound on the achievable code length, and hence is a useful tool to evaluate the performance of existing heuristic context quantizers.

Context based entropy coding often faces the conflict of a desire for large templates and the problem of context dilution. We consider the problem of finding the quantizer Q that quantizes the K-dimensional causal context C/sub i/=(X(i-t/sub 1/), X(i-t/sub 2/), ..., X(i-t/sub K/)) of a source symbol X/sub i/ into one of M conditioning states. A solution giving the minimum adaptive code length for a given data set is presented (when the cost of the context quantizer is neglected). The resulting context quantizers can be used for sequential coding of the sequence X/sub 0/, X/sub 1/, X/sub 2/, .... A coding scheme based on binary decomposition and context quantization for coding the binary decisions is presented and applied to digital maps and /spl alpha/-plane sequences. The optimal context quantization is also used to evaluate existing heuristic context quantizations.

If the calculation of the critical length is based on the expurgated exponent, the length becomes nonzero for low error probabilities. This result applies to typical long codes, but it may also be useful for modeling error events in specific codes.