In this paper, the existential lower bounds on asymptotic distance ratios of error-correcting codes with J(J2) levels of unequal error protection given by encoding and decoding methods based on the concept of the generalized concatenated codes are obtained. The constructive concatenated codes with unequal error protection are proposed for high and low code rates. The formaer is constructed by the generalized version of Justesen codes. The latter is constructed by the secondorder concatenation with the generalized concatenated codes. Comparing the existential lower bounds on asymptotic distance ratios with those of the constructive concatenated codes, the lower bounds of the constructive codes are lower than the existential lower bounds.

Trade-offs between storage and error in the simple binary-valued retrieval problems are analyzed by using the rate distortion theoretic method from the viewpoint of the trade-offs evaluation model. These problems, the table lookup question, the exact match question, and the set of all binary-valued questions, have been proposed as the typical basic model of the information retrieval systems to evaluate the memory cost, the access cost, the state cost, the logic cost, and their relationships. The evaluation critera of memory-error trade-offs are the elastic condition and the excess of information. The former means that drastic savings of the amount of storage with a small error tolerance is feasible. The latter means the measure to evaluate the degree of the possibility for the system to achieve elastic condition. As a result, although the set of all binary-valued questions admits the possible highest excess of information, its statistical property is almost equivalent to that of the table lookup question, and only the exact match question satisfies the elastic condition. Furthermore, considering the conjunctive model which means the combination of each questions for the basic model, the property of elasticity or inelasticity for the basic model is changed from that of the basic model in accordance with the degree of the combination of the questions.

In this paper, the criterion for evaluating inductive inference is considered from the information theoretical concept, especially from the view points of source coding and decision theory. Although the finite behavior of the inductive inference methods is important in the application field, there is no theoretical criterion for evaluating hypotheses at a finite stage of process. In the previous paper, we have defined an amount of semantic information included in well-formed formulas (wff) and denoted an analogy between inference and source coding. Inductive inference is regarded as source encoding, assuming that observed facts are compressed into a hypothesis similar to the way a source sequence is compressed into a codeword. By using the above definition, we can apply the criterion of source coding especially Minimum Description Length (MDL) to inductive inference. Therefore, the description length for representing a hypothesis is suitable for the criterion for evaluating a hypothesis inference at the finite stage of the inductive inference process. Moreover, by using the relation between the MDL criterion and Bayes risk, inductive inference can be interpreted with Bayes decision theory. The algorithm for the inductive inference of Horn clause is shown as an application of the proposed criterion. In the algorithm, the search space of hypotheses is restricted by using a refinement relation for increasing efficiency.

The original iterated codes proposed by P. Elias can be regarded as the codes constructed by iterating two dimensional product codes. While the modified product codes have been proposed and shown to be able to increase the rates without increasing the probability of decoding error. On the other hand, we have proposed new codes, called modified iterated codes A, and improved the performance for the original iterated codes by applying the coding and the decoding schemes of the modified product codes to these product codes. It has been proved that the rates of codes A were always much higher than those of the original iterated codes for cross-over probability p0.1617. In this paper, by additionally combining the coding and the decoding schemes of the superimposed codes constructed on the basis of the modified product codes, which are able to increase the rates with the same minimum distance, the performance for codes A can be further improved. We call codes thus constructed modified iterated codes B. It is shown that the rates of codes B are partially higher than those of codes A for p0.0959.