Knowledge representation is one of the important topics in data mining research. In this paper, based on the feature space theory in data mining, the transformation between extensions and intensions of concepts is discussed in detail. First, inner projections of fuzzy relations, as a basic mathematical tool, are defined, and properties of inner projections are discussed. Then inner transformation of fuzzy relations, inverse inner transformations, and related properties are introduced. The concept structure is shown by feature spaces. Lastly, transformations between extensions and intensions are discussed.

A trellis code encoded by using the encoder of a convolutional code C with a short constraint length followed by an additional processing unit is equivalent to a trellis code with a large constraint-length. In 1993, Hellstern proposed a trellis coding scheme for which the processing unit consists of a delay processor and a signal mapper. With Hellstern's scheme, trellis codes with large free distances can be constructed. In this paper, we propose two trellis coding schemes. For the first scheme, the processing unit is composed of multiple pairs of delay processors and signal mappers. For the second scheme, the processing unit is composed of a convolutional processor and a signal mapper, where a convolutional processor is a rate 1 convolutional code. The trellis code constructed from each of the proposed schemes can be suboptimally decoded by using the trellis of the convolutional code C with some feedback information. Either of the proposed schemes can produce a trellis code that has a larger bound on free distance and better error performance as compared to the trellis code constructed from Hellstern's scheme based on the same convolutional code C.

Block-coded modulation with interblock memory (BCMIM) is a variation of block-coded modulation (BCM) which is designed for multilevel coding. By providing interblock memory between adjacent blocks, the coding rate of a BCMIM scheme can be increased without decreasing the minimum squared Euclidean distance (MSED) as compared to the original BCM. In an early version of BCMIM, interblock coding is provided only between the first two coding levels of adjacent blocks. In this paper, we design BCMIM with a more general form for which interblock coding can be introduced among many coding levels. In this way, we can further increase the coding rate of BCMIM without decreasing the MSED. We provide many examples to show the advantages of BCMIM with the general form. Most of the examples are designed based on multidimensional signal sets, since a multidimensional signal set can provide more coding levels than a two-dimensional (2-D) signal set.

We propose a trellis-coding scheme for which large free distances and low decoding complexities can be easily achieved. The proposed trellis-coding scheme is a generalization of the trellis-coding scheme proposed by Hellstern (1993). With the proposed trellis-coding scheme, we have the flexibility of controlling the decoding delay.

We propose a very powerful trellis coded modulation (TCM) scheme which has the structure of multilevel and single-stage coding.For the proposed TCM scheme, the encoder consists of a binary convolutional encoder which is followed by a multilevel delay processor and a signal mapper.By computer search, we find many useful TCM systems which are good for AWGN channels.