Nowadays, error control codes have become an essential technique to improve the reliability of various digital systems. A new type error control codes called m-spotty byte error control codes are applied to computer memory systems. These codes are essential to make the memory systems reliable. Here, we introduce the m-spotty Rosenbloom-Tsfasman weights and m-spotty Rosenbloom-Tsfasman weight enumerator of linear codes over Fq[u]/(uk) with uk=0. We also derive a MacWilliams type identity for m-spotty Rosenbloom-Tsfasman weight enumerator.

Granular gain of low-dimensional lattices based on binary linear codes is estimated using a quantization algorithm which is equivalently a soft-decision decoding of the underlying code. It is shown that substantial portion of the ultimate granular gain is achieved even in limited dimensions.

An important concept in secret sharing scheme is the access structure. However, determining the access structure of the secret sharing scheme based on a linear code is a very difficult problem. In this work, we provide a method to construct a class of two-weight linear codes over finite rings. Based on the two-weight codes, we present an access structure of a secret sharing scheme.

Recently, Haley and Grant introduced the concept of reversible codes – a class of binary linear codes that can reuse the decoder architecture as the encoder and encodable by the iterative message-passing algorithm based on the Jacobi method over F2. They also developed a procedure to construct parity check matrices of a class of reversible codes named type-I reversible codes by utilizing properties specific to circulant matrices. In this paper, we refine a mathematical framework for reversible codes based on circulant matrices through a ring theoretic approach. This approach enables us to clarify the necessary and sufficient condition on which type-I reversible codes exist. Moreover, a systematic procedure to construct all circulant matrices that constitute parity check matrices of type-I reversible codes is also presented.

Low-density parity-check (LDPC) codes become very popular in channel coding, since they can achieve the performance close to maximum-likelihood (ML) decoding with linear complexity of the block length. Recently, Muramatsu et al. proposed a code using LDPC matrices for Slepian-Wolf source coding, and showed that their code can achieve any point in the achievable rate region of Slepian-Wolf source coding. However, since they employed ML decoding, their decoder needs to know the probability distribution of the source. Hence, it is an open problem whether there exists a universal code using LDPC matrices, where universal code means that the error probability of the code vanishes as the block length tends to infinity for all sources whose achievable rate region contains the rate pair of encoders. In this paper, we show the existence of a universal Slepian-Wolf source code using LDPC matrices for stationary memoryless sources.

In digital signal processing, the sampling theorem states that any real valued function f can be reconstructed from a sequence of values of f that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of f. This theorem can also be applied to functions over finite domain. Then, the range of frequencies of f can be expressed in more detail by using a bounded set instead of the maximum frequency. A function whose range of frequencies is confined to a bounded set is referred to as bandlimited function. And a sampling theorem for bandlimited functions over Boolean domain has been obtained. Here, it is important to obtain a sampling theorem for bandlimited functions not only over Boolean domain (GF(2)n domain) but also over GF(q)n domain, where q is a prime power and GF(q) is Galois field of order q. For example, in experimental designs, although the model can be expressed as a linear combination of the Fourier basis functions and the levels of each factor can be represented by GF(q), the number of levels often take a value greater than two. However, the sampling theorem for bandlimited functions over GF(q)n domain has not been obtained. On the other hand, the sampling points are closely related to the codewords of a linear code. However, the relation between the parity check matrix of a linear code and any distinct error vectors has not been obtained, although it is necessary for understanding the meaning of the sampling theorem for bandlimited functions. In this paper, we generalize the sampling theorem for bandlimited functions over Boolean domain to a sampling theorem for bandlimited functions over GF(q)n domain. We also present a theorem for the relation between the parity check matrix of a linear code and any distinct error vectors. Lastly, we clarify the relation between the sampling theorem for functions over GF(q)n domain and linear codes.

This paper clarifies the adequacy of the linear channel coding approach for the source coding with partial side information at the decoder. A sufficient condition for an ensemble of linear codes which achieves the Wyner's bound is given. Our result reveals that, by combining a good lossy code, an LDPC code ensemble gives a good code for source coding with partial side information at the decoder.

This paper deals with a secret key agreement problem from correlated random numbers. It is proved that there is a pair of linear matrices that yields a secret key agreement in the situation wherein a sender, a legitimate receiver, and an eavesdropper have access to correlated random numbers. A relation between the coding problem of correlated sources and a secret key agreement problem from correlated random numbers are also discussed.

Orthogonal Arrays (OAs) have been playing important roles in the field of experimental design. It has been known that OAs are closely related to error-correcting codes. Therefore, many OAs can be constructed from error-correcting codes. But these OAs are suitable for only cases that equal interaction effects can be assumed, for example, all two-factor interaction effects. Since these cases are rare in experimental design, we cannot say that OAs from error-correcting codes are practical. In this paper, we define OAs with unequal strength. In terms of our terminology, OAs from error-correcting codes are OAs with equal strength. We show that OAs with unequal strength are closer to practical OAs than OAs with equal strength. And we clarify the relation between OAs with unequal strength and unequal error-correcting codes. Finally, we propose some construction methods of OAs with unequal strength from unequal error-correcting codes.

The explicit construction of a universal source code for correlated sources is presented. The construction is based on a technique of simulated random coding algorithms [5]. The proposed algorithm simulates the random generation of linear codes. For every pair of correlated sources whose achievable rate region includes a given pair of encoding rates, the decoding error rate of the proposed algorithm goes to zero almost surely as the block length goes to infinity.

Probabilistic encryption becomes more and more important since its ability to against chosen-ciphertext attack. Applications like online voting schemes and one-show credentials are based on probabilistic encryption. Research on good probabilistic encryptions are on going, while many good deterministic encryption schemes are already well implemented and available in many systems. To convert any deterministic encryption scheme into a probabilistic encryption scheme, a randomized media is needed to apply on the message and carry the message over as an randomized input. In this paper, nonlinear codes obtained by certain mapping from linear error-correcting codes are considered to serve as such carrying media. Binary nonlinear codes obtained by Gray mapping from 4-linear codes are discussed as example for a such scheme.

Recently, Vatan, Roychowdhury and Anantram have presented two types of revised versions of the Calderbank-Shor-Steane code construction, and have also provided an exhaustive procedure for determining bases of quantum error-correcting codes. In this paper, we investigate the revised versions given by Vatan et al., and point out that there is no essential difference between them. In addition, we propose an efficient algorithm for searching for bases of quantum error-correcting codes. The proposed algorithm is based on some fundamental properties of classical linear codes, and has much lower complexity than Vatan et al.'s procedure.

The minimum distance of a linear code C is a useful metric property for estimating the error correction upper bound of C and the maximum likelihood decoding of a linear code C is also of practical importance and of theoretical interest. These problems are known to be NP-hard to approximate within any constant relative error to the optimum. As a problem related to the above, we consider the maximization problem MAX-WEIGHT: Given a generator matrix of a linear code C, find a codeword c C with its weight as close to the maximum weight of C as possible. It is shown that MAX-WEIGHT PTAS unless P=NP, however, no nontrivial approximation upper and lower bounds are known. In this paper, we investigate the complexity of MAX-WEIGHT to make the approximation upper and lower bounds more precise, and show that (1) MAX-WEIGHT is APX-complete; (2) MAX-WEIGHT is approximable within relative error 1/2 to the optimum; (3) MAX-WEIGHT is not approximable within relative error 1/10 to the optimum unless P=NP.

Motivated by the design of fault-tolerant multiprocessor interconnection networks, this paper considers the following problem: Given a positive integer t and a graph H, construct a graph G from H by adding a minimum number Δ(t, H) of edges such that even after deleting any t edges from G the remaining graph contains H as a subgraph. We estimate Δ(t, H) for the hypercube and torus, which are well-known as important interconnection networks for multiprocessor systems. If we denote the hypercube and the square torus on N vertices by QN and DN respectively, we show, among others, that Δ(t, QN) = O(tN log(log N/t + log 2e)) for any t and N (t 2), and Δ(1, DN) = N/2 for N even.

McEliece and Swanson offerred an upper bound on the decorder error probability of Reed-Solomon codes. In this paper, we investigate the decorder error probability of binary linear block codes and verify its properties, and apply it to binary primitive BCH codes. It is shown that the decorder error probability of an (n,k,t) binary linear block code is determined by PE uniquely if it is a constant. We derive the decorder error probability of (n,k,t) binary primitive BCH codes with n=2m-1 and +1 and show that the decorder error probabilities of those codes are close to PE if codelengh is large and coderate is high. We also compute and analyze the decorder error probabilities of some binary primitive BCH codes.