Applications of Graph Theory towards Data Storage Media

Graph theory, which studies the theoretical properties of graphs, is an area of applied and discrete mathematics, and has various applications towards engineering. For example, network topologies can be characterized by graphs and those topologies can be used to find better solutions in smooth communications. In this paper, we explain how graph theory has been applied in data storage media. In particular, we explain storing/reading schemes for typical data storage media, and then present how graph theory plays an important role in coding for data storage media, together with our contributions. More precisely, we review our results from constrained systems, which control the appearance of certain data sequences, and how the results work for reliable data storage media

A Universal Two-Dimensional Source Coding by Means of Subblock Enumeration

The technique of lossless compression via substring enumeration (CSE) is a kind of enumerative code and uses a probabilistic model built from the circular string of an input source for encoding a one-dimensional (1D) source. CSE is applicable to two-dimensional (2D) sources, such as images, by dealing with a line of pixels of a 2D source as a symbol of an extended alphabet. At the initial step of CSE encoding process, we need to output the number of occurrences of all symbols of the extended alphabet, so that the time complexity increases exponentially when the size of source becomes large. To reduce computational time, we can rearrange pixels of a 2D source into a 1D source string along a space-filling curve like a Hilbert curve. However, information on adjacent cells in a 2D source may be lost in the conversion. To reduce the time complexity and compress a 2D source without converting to a 1D source, we propose a new CSE which can encode a 2D source in a block-by-block fashion instead of in a line-by-line fashion. The proposed algorithm uses the flat torus of an input 2D source as a probabilistic model instead of the circular string of the source. Moreover, we prove the asymptotic optimality of the proposed algorithm for 2D general sources

Compression by Substring Enumeration Using Sorted Contingency Tables

This paper proposes two variants of improved Compression by Substring Enumeration (CSE) with a finite alphabet. In previous studies on CSE, an encoder utilizes inequalities which evaluate the number of occurrences of a substring or a minimal forbidden word (MFW) to be encoded. The inequalities are derived from a contingency table including the number of occurrences of a substring or an MFW. Moreover, codeword length of a substring and an MFW grows with the difference between the upper and lower bounds deduced from the inequalities, however the lower bound is not tight. Therefore, we derive a new tight lower bound based on the contingency table and consequently propose a new CSE algorithm using the new inequality. We also propose a new encoding order of substrings and MFWs based on a sorted contingency table such that both its row and column marginal total are sorted in descending order instead of a lexicographical order used in previous studies. We then propose a new CSE algorithm which is the first proposed CSE algorithm using the new encoding order. Experimental results show that compression ratios of all files of the Calgary corpus in the proposed algorithms are better than those of a previous study on CSE with a finite alphabet. Moreover, compression ratios under the second proposed CSE get better than or equal to that under a well-known compressor for 11 files amongst 14 files in the corpus

六角形格子上の積符号を用いた符号化変調方式によるPAPRの低減

小形通信機器を用いる無線通信においては，寸法や重量が制限されるため，より電力効率の良い通信方式が求められる．本論文では，電力利用効率の良い通信方式として，六角形格子上の整数符号を用いた積符号による符号化変調方式を提案する．六角形格子は信号点の密度が高いため，等方性の19点六角形格子のピーク対平均電力比は矩形16値直交振幅変調(QAM: Quadrature Amplitude Modulation)よりも0.56dB低く抑えることができる．また，整数符号は，訂正可能な誤りを近接点に設定でき，遠方の信号点へ誤る確率よりも近接点へ誤る確率の方が高いと言う実際的な通信路に適している．本検討では，19，37及び61点の六角形格子上で整数符号単体及び整数符号と負巡回符号またはリード・ソロモン符号の積符号のシミュレーションを行った．その結果，整数符号とリード・ソロモン符号の積符号が，負巡回符号との積符号や各符号単体の結果と比べてビット誤り率(BER: Bit Error Rate)の改善効果が大きいことを示すことができた．また，信号点数の近い矩形QAMと比較してBERが1.0×10-5において0.8dBから1dBの符号化利得を得ることができ，提案符号化変調方式による電力及びBERの改善効果を確認できた．Wireless communication using small devices requires high power efficiency schemes because of size and weight constraints. In this paper, coded modulation schemes using product codes composed of integer codes over hexagonal constellations are proposed. The peak-to-average power ratio of an isotropy 19-Hexagonal constellation is 0.56 dB lower than that of 16QAM. In addition, integer codes can define its correctable error in the vicinity of signal points. Therefore, it is suitable for the practical communication channel. Our coded modulation simulation results over 19, 37 and 61 hexagonal constellations showed lower bit error rate than the results on square QAM constellations