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.

According to the properties found in the algebraic complete decoding method for triple-error-correcting binary Bose-Chaudhuri-Hocquenghem (BCH) codes, a step-by-step complete decoding algorithm of this code is presented.

In this letter, using techniques from linear algebra and coding theory, we characterize the quadratic Boolean functions represented by trace. We show that a linear combination of trace-terms over finite field can be determined to be bent by a polynomial GCD computation. Then we derive some new families of bent functions.

Cryptography and Coding Theory are closely related in many respects. Recently, the problem of "decoding Reed Solomon codes" (also known as "polynomial reconstruction") was suggested as an intractability assumption to base the security of protocols on. This has initiated a line of cryptographic research exploiting the rich algebraic structure of the problem and its variants. In this paper we give a short overview of the recent works in this area as well as list directions and open problems in Polynomial Reconstruction Based Cryptography.

Almost all of the current public-key cryptosystems (PKCs) are based on number theory, such as the integer factoring problem and the discrete logarithm problem (which will be solved in polynomial-time after the emergence of quantum computers). While the McEliece PKC is based on another theory, i.e. coding theory, it is vulnerable against several practical attacks. In this paper, we summarize currently known attacks to the McEliece PKC, and then point out that, without any decryption oracles or any partial knowledge on the plaintext of the challenge ciphertext, no polynomial-time algorithm is known for inverting the McEliece PKC whose parameters are carefully chosen. Under the assumption that this inverting problem is hard, we propose a slightly modified version of McEliece PKC that can be proven, in the random oracle model, to be semantically secure against adaptive chosen-ciphertext attacks. Our conversion can achieve the reduction of the redundant data down to 1/3-1/4 compared with the generic conversions for practical parameters.

This paper suggests modified LZSS which is suitable for compressing Hangul data by Hangul character token and the string token with small size based on Hangul properties. The Hangul properties can be described in 2 ways. 1) The structure of a Hangul character consists of 3 letters: The first sound letter, the middle sound letter, and the last sound letter which are called Cho-seong, Jung-seong, and Jong-seong, respectively. 2) The code of Hangul is represented by 2 bytes. The first property is used for making the character token processing Hangul characters which occupies most of the unmatched characters. That is, the unmatched Hangul characters are replaced with one Hangul character token represented by Huffman codes of Cho-seong, Jung-seong, and Jong-seong in regular sequence, instead of 2 character tokens. The second property is used to shorten the size of the string token processing matched string. In other words, since more than 75% of Hangul data are Hangul and Hangul codes are constructed in 2 bytes, the addresses of the window of LZSS can be assigned in 2-byte unit. As a result, the distance field and the length field of the string token can be lessened by one bit each. After compressing Hangul data through these tokens, about 3% of improvement could be made in compression ratio.

In the band-limited signal space, the mutual relation between continuous time signal and discrete time signal is expressed by the sampling theorem of Whittaker-Someya-Shannon. This theorem consists of an orthonormal expansion formula using sinc functions. In that formula, the expansion coefficients are identical to the sample values of signals. In general, the bandlimited signal space is not always suited to model the signals in nature. The authors have proposed a new model for signal processing based on finite times continuously differentiable functions. This paper aims to complete the sampling theorem for the spline signal spaces, which corresponds to the sampling theorem of Whittaker-Someya-Shannon in the band-limited signal space. Since the obtained sampling theorem gives the simplest representation of signals, it is considered to be the most fundamental characterization of spline functions used for signal processing. The biorthonormal basis derived in this paper is considered to be a system of delta functions at sampling points with some continuous differentiability.

This paper presents an adaptive rate error control scheme for digital communication over time-varying channels. The cyclic code with majority-logic decoding is used in a cascaded way as an inner code to create a simple and powerful hybrid-ARQ error control scheme. Inner code is used only for error correction and the outer code is used for both error correction and error detection. When an error is detected, retransmission is required. The unsuccessful packets are not discarded as with conventional schemes, but are combined with their retransmitted copies. Approximations for the throughput efficiency and the undetectable error probability are given. A high reliability coupled with a simple high-speed implementation makes it suitable for high data rate error control over both stationary and nonstationary channels. Adaptive error control scheme becomes the best solution for time-varying channels when the optimum code is selected according to the actual channel conditions to enhance the system performance. The main feature of this system is that the basic structure of the encoder and decoder need not be modified while the error-correction capability of the code increases. Results of a comparative analysis show that the proposed scheme outperforms other similar ARQ protocols.

A technique is presented for constructing (d,k) block codes capable of detecting single bit errors and single peak-shift errors in consecutive two runs. This constrains the runlengths in the code sequences to odd numbers. The capacities and the cardinalities for finite code length of these codes are described. A technique is also proposed for constructing (d,k) block codes capable of correcting single peak-shift errors.

In this paper a recursive form of Welch-Berlekamp (W-B) algorithm is provided which is a novel and fast decoding algorithm.

In this letter, we consider a magnetic or optical recording system employing a concatenated code that consists of a runlength-limited (d, k) block code as an inner code and a byte-error-correcting code as an outer code. (d, k) means that any two consecutive ones in the code bit stream are separated by at least d zeros and by at most k zeros. The minimum separation d and the maximum separation k are imposed in order to reduce intersymbol interference and extract clock control from the received bit stream, respectively. This letter recommends to use as the outer code a unidirectional-byte-error-correcting code instead of an ordinary byte-error-correcting code. If we devise the mapping of the code symbols of the outer code onto the codewords of the inner code, we may improve the error performance. Examples of the mappings are described.

An Interlace Coding System (ICS) involving data compression code, data encryption code and error correcting code is proposed and its error performance on additive white Gaussian noise (AWGN) channel with quadrature phase shift keying (QPSK) is analyzed. The proposed system handles data compression, data encryption and error correcting processes together, i.e. adds error correcting redundancy to the block lists of the dictionary in which compression system constructs to reduce source redundancy. Each block list is encoded by Ziv-Lempel code and Data Encryption Standard (DES). As the catastrophic condition determined by the data compression procedure is not negligible, error correcting redundancy should be added so as to avoid catastrophic condition. We found that the catastrophic condition depends only on the size of the dictionary for our proposed system. Thus, by employing a large dictionary, good error performance can be applied by the proposed system and the catastrophic condition can be avoided.

A new class of (m23m1,m2) 1-step majority-logic decodable double error correcting codes (1-step DEC codes) is described, where m is an odd integer. Combining this code with properly constructed (m1k1,k1) and (m,k2) 1-step DEC codes, a (m23(mk1)1,m23k1) 1-step DEC code and a (m23(mk2)1,m2) 2-step majority-logic decodable DEC code (2-step DEC code) are obtained, respectively. Considering computer memory applications, some practical 1 -and 2-step DEC codes with data-bit lengths of 24, 32, 64 and 72 are obtained by shortening the new codes, and are compared to existing majority-logic decodable DEC codes. It is shown that, for given data-bit lengths, new 2-step DEC codes have much better code rates than self-orthogonal DEC codes but slightly worse code rates than existing 2-step majority-logic decodable cyclic DEC codes (2-step cyclic DEC codes). However, parallel decoders of new 2-step DEC codes are much simpler than those of exisiting 2-step cyclic DEC codes, and are nearly as simple as those of 1-step DEC codes.