Train marshalling is the process of reordering the railcars of a train in such a way that the railcars with the same destination appear consecutively in the final, reassembled train. The process takes place in the shunting yard by means of a number of classification tracks. In the train marshalling problem (TMP), the objective is to perform this rearrangement of the railcars with the use of as few classification tracks as possible. The problem has been shown to be NP-hard, and several exact and approximation algorithms have been developed for it. In this paper, we propose a novel exact dynamic programming (DP) algorithm for the TMP. The worst-case time complexity of this algorithm (which is exponential in the number of destinations and linear in the number of railcars) is lower than that of the best presently available algorithm for the problem, which is an inclusion-exclusion-based DP algorithm. In practice, the proposed algorithm can provide a substantially improved performance compared to its inclusion-exclusion-based counterpart, as demonstrated by the experimental results.

Error-rate floor phenomenon is known to be a serious impediment to the use of low-density parity-check (LDPC) codes for some practical applications that demand high data reliability. In the case of binary erasure channels (BECs), certain error-prone patterns, known as stopping sets, are proven to cause this performance degradation. A possible approach to diminish this drawback over BECs is to eliminate stopping sets by parity-check matrix extension. Given a parity-check matrix H, and a list L of its stopping sets, we present an integer linear programming (ILP) formulation to find a parity-check equation which eliminates the maximum number of stopping sets in L. One of the distinguishing advantages of the proposed scheme is its flexibility for modifications such as: limiting the weight of the new parity-check row, making the new row redundant or linearly independent, 4-cycle avoidance, and taking into account the sizes of stopping sets. Armed with these adjustments, the method can provide good performance improvements, as evidenced by simulation results. Furthermore, for a given Q ∈ N, by extending the basic formulation, we provide an ILP formulation for finding a set of size Q of parity-check equations which can best eliminate the stopping sets in L, among all such sets.

A new combinatorial technique for constructing girth-6 structured binary regular low-density parity-check (LDPC) codes based on special types of t-designs is given. A very large number of well-known t-designs can be used by this method for code construction. Based on this method, a t-(v,k,λ) design D=(X,B) can be exploited for code construction if it satisfies the following three conditions: 1) |B1 ∩ B2|≤ t for any two blocks B1,B2∈ B and B1≠ B2; 2) λ>1; and 3) k>t. Though the technique works for any t-design satisfying these conditions, we focus only on the utilization of simple triple systems, super-simple BIBDs, Steiner systems, and large sets (LSs) of t-designs. We also construct binary and non-binary girth-6 QC-LDPC codes from the t-designs satisfying these conditions by using matrix dispersion method. Experimental results show that the constructed non-binary QC-LDPC codes can provide good practical performance under iterative decoding using the fast Fourier transform based q-ary sum-product algorithm (FFT-QSPA) and they can achieve acceptable coding gains over random-like codes of comparable parameters decoded with sum-product algorithm (SPA).

This paper presents a novel method for constructing structured regular low-density parity-check (LDPC) codes based on a special type of combinatorial designs, known as Steiner systems. This code design approach can be considered as a generalization of the well-known method which uses the point-block incidence matrix of a Steiner 2-design for the code construction. Though the given method can be applied on any Steiner system S(t, k, v), in this paper we focus only on Steiner systems with t ≥ 3. Furthermore, we show that not only a Steiner system (X,B) itself, but also its residual design with respect to an arbitrary point x ∈ X can be employed for code construction. We also present a technique for constructing binary and non-binary QC-LDPC codes from Steiner systems. The Tanner graph of the constructed codes is free of 4-cycles and hence the codes have girth at least six. Simulation results show that the so constructed codes perform well over the AWGN channel with iterative message-passing decoding.