Feedback Communication Systems with Limitations on Incremental Redundancy

This paper explores feedback systems using incremental redundancy (IR) with noiseless transmitter confirmation (NTC). For IR-NTC systems based on {\em finite-length} codes (with blocklength $N$) and decoding attempts only at {\em certain specified decoding times}, this paper presents the asymptotic expansion achieved by random coding, provides rate-compatible sphere-packing (RCSP) performance approximations, and presents simulation results of tail-biting convolutional codes. The information-theoretic analysis shows that values of $N$ relatively close to the expected latency yield the same random-coding achievability expansion as with $N = \infty$. However, the penalty introduced in the expansion by limiting decoding times is linear in the interval between decoding times. For binary symmetric channels, the RCSP approximation provides an efficiently-computed approximation of performance that shows excellent agreement with a family of rate-compatible, tail-biting convolutional codes in the short-latency regime. For the additive white Gaussian noise channel, bounded-distance decoding simplifies the computation of the marginal RCSP approximation and produces similar results as analysis based on maximum-likelihood decoding for latencies greater than 200. The efficiency of the marginal RCSP approximation facilitates optimization of the lengths of incremental transmissions when the number of incremental transmissions is constrained to be small or the length of the incremental transmissions is constrained to be uniform after the first transmission. Finally, an RCSP-based decoding error trajectory is introduced that provides target error rates for the design of rate-compatible code families for use in feedback communication systems.Comment: 23 pages, 15 figure

Stopping Set Distributions of Some Linear Codes

Stopping sets and stopping set distribution of an low-density parity-check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let $C$ be a binary $[n,k]$ linear code with parity-check matrix $H$, where the rows of $H$ may be dependent. A stopping set $S$ of $C$ with parity-check matrix $H$ is a subset of column indices of $H$ such that the restriction of $H$ to $S$ does not contain a row of weight one. The stopping set distribution $\{T_i(H)\}_{i=0}^n$ enumerates the number of stopping sets with size $i$ of $C$ with parity-check matrix $H$. Note that stopping sets and stopping set distribution are related to the parity-check matrix $H$ of $C$. Let $H^{*}$ be the parity-check matrix of $C$ which is formed by all the non-zero codewords of its dual code $C^{\perp}$. A parity-check matrix $H$ is called BEC-optimal if $T_i(H)=T_i(H^*), i=0,1,..., n$ and $H$ has the smallest number of rows. On the BEC, iterative decoder of $C$ with BEC-optimal parity-check matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes and the extended Hamming codes.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201

Optimization of Parity-Check Matrices of LDPC Codes

Madala tihedusega paarsuskontroll (LDPC) on laialdaselt kasutusel kommunikatsioonis tänu oma suurepärasele praktilisele võimekusele. LDPC koodi vigade tõenäosust iteratiivse dekodeerimise puhul binaarsel kustutuskanalil määrab klass kombinatoorseid objekte, nimega peatamise rühm. Väikese suurusega peatamise rühmad on dekodeerija vigade põhjuseks. Peatamise liiasust määratletakse kui minimaalset ridade arvu paarsuskontrolli koodi maatriksis, mille puhul pole selles väikesi peatuse rühmi. Han, Siegel ja Vardy kasutavad üld binaarse lineaarkoodi ülemise piiri peatamiste liiasuse tuletamiseks tõenäosuslikku analüüsi. Need piirid on teadaolevalt parimad paljude koodi perekondade puhul. Selles töös me parendame Hani, Siegeli ja Vardy tulemusi modifitseerides selleks nende analüüsi. Meie lähenemine erineb sellepoolest, et me valime mõistlikult esimese ja teise rea paarsuskontrolli maatriksis ja siis läheme edasi tõenäosusliku analüüsiga. Numbrilised väärtused kinnitavad seda, et piirid mis on määratletud selles töös on paremad Hani, Siegeli ja Vardy omadest kahe koodi puhul: laiendatud Golay koodis ja kvadraatses jääk koodis pikkusega 48.Low-density parity-check (LDPC) codes are widely used in communications due to their excellent practical performance. Error probability of LDPC code under iterative decoding on the binary erasure channel is determined by a class of combinatorial objects, called stopping sets. Stopping sets of small size are the reason for the decoder failures. Stopping redundancy is defined as the minimum number of rows in a parity-check matrix of the code, such that there are no small stopping sets in it. Han, Siegel and Vardy derive upper bounds on the stopping redundancy of general binary linear codes by using probabilistic analysis. For many families of codes, these bounds are the best currently known. In this work, we improve on the results of Han, Siegel and Vardy by modifying their analysis. Our approach is different in that we judiciously select the first and the second rows in the parity-check matrix, and then proceed with the probabilistic analysis. Numerical experiments confirm that the bounds obtained in this thesis are superior to those of Han, Siegel and Vardy for two codes: the extended Golay code and the quadratic residue code of length 48