Class Of Binary Codes Research Articles

Value functions and two classes of binary codes

We state that the minimum and the maximum support weight of subcodes of a code are useful in several aspects. Based on the above findings, we will determine part of the above mentioned parameters for two classes of binary codes. We also present the separation property and calculate the trellis complexity of a class of binary codes.

On Cyclic Codes of Odd Lengths from the Stable Variety of Regular Cayley Graphs

The use of the adjacency matrix of a graph as a generator matrix for some classes of binary codes had been reported and studied. This paper concerns the utilization of the stable variety of Cayley regular graphs of odd order for efficient interconnection networks as studied, in the area of Codes Generation and Analysis. The Use of some succession scheme in the construction of a stable variety of the Cayley regular graph had been considered. We shall enumerate the adjacency matrices of the regular Cayley graphs so constructed which are of odd order (2m+1), for m≥3 as in [1]. Next, we would show that the Matrices are cyclic and can be used in the generation of cyclic codes of odd lengths.

On the intersection of binary linear codes

For a binary linear code, a new relation between the intersection and (2, 2)-separating property is addressed, and a relation between the intersection and the trellis complexity is also given. Using above relations, the authors will apply several classes of binary codes to secret sharing scheme and determine their trellis complexity and separating properties. The authors also present the properties of the intersection of certain kinds of two-weight binary codes. By using the concept of value function, the intersecting properties of general binary codes are described.

Reed-Muller Codec Simulation Performance

The approach to error correction coding taken by modern digital communication systems started in the late 1940's with the ground breaking work of Shannon, Hamming and Golay. Reed- Muller (RM) codes were an important step beyond the Hamming and Golay codes because they allowed more flexibility in the size of the code word and the number of correctable errors per code word. Whereas the Hamming and Golay codes were specific codes with particular values for q; n; k; and t, the RM codes were a class of binary codes with a wide range of allowable design parameters. Binary Reed-Muller codes are among the most prominent families of codes in coding theory. They have been extensively studied and employed for practical applications. In this research, the performance simulation of Reed-Muller Codec was presented. An introduction on Reed-Muller codes, were introduced that consists of defining the key terms and operation used with the binary numbers. Reed-Muller codes were defined and encoding matrices were discussed. The decoding process was given and some examples were demonstrated to clarify the method. The results and the performance of Reed-Muller encoding were presented and the messages been encoded using the defined matrices were shown. The simulation of the decoding part also been shown. The performance of Reed-Muller codes were then analyzed in terms of its code rate, code length and minimum Hamming distance. The analysis that performed also successfully examines the relationship between the parameters of Reed- Muller coding. The decoding part of the Reed-Muller codes can detect one error and correct it as shown in the examples.

On a Class of Constant Weight Codes

For any odd prime power $q$ we first construct a certain non-linear binary code $C(q,2)$ having $(q^2-q)/2$ codewords of length $q$ and weight $(q-1)/2$ each, for which the Hamming distance between any two distinct codewords is in the range $[q/2-3\sqrt q/2,\ q/2+3\sqrt q/2]$ that is, 'almost constant'. Moreover, we prove that $C(q,2)$ is distance-invariant. Several variations and improvements on this theme are then pursued. Thus, we produce other classes of binary codes $C(q,n)$, $n\geq 3$, of length $q$ that have 'almost constant' weights and distances, and which, for fixed $n$ and big $q$, have asymptotically $q^n/n$ codewords. Then we prove the possibility of extending our codes by adding the complements of their codewords. Also, by using results on Artin $L-$series, it is shown that the distribution ofthe $0$'s and $1$'s in the codewords we constructed is quasi-random. Our construction uses character sums associated with the quadratic character $\chi$ of $F_{q^n}$ in which the range of summation is $F_q$. Relations with the duals of the double error correcting BCH codes and the duals of the Melas codes are also discussed.

Relational valued fuzzy sets

A generalization of the notion of a fuzzy set is given. An R-fuzzy set A is a mapping from a set A into the relational system ( S, ϱ) where ϱ is a binary relation on S. Necessary and sufficient conditions for a unique decomposition and synthesis of an R-fuzzy set into the family of ordinary subsets ( p-cuts) are given. As a consequence, it is possible to obtain a fuzzy set as a synthesis of any collection of characteristics functions on a nonvoid set, which was impossible in classical fuzzy set theory. As a contribution to coding theory, a possibility to express any binary block code syntheticly, by a single fuzzy set, is obtained. Note that up to now only some classes of binary codes have been characterized by fuzzy sets. Suitable examples (BCD and a linear code) are given at the end of the paper, together with necessary and sufficient conditions under which an R-valued fuzzy set corresponds to a linear code.

New class of double inductive binary error correcting codes based on the class of Reed–Muller codes

A new class of binary codes is introduced. The codes have length n = 1.5 · 2m (m ≥ 3), and have a similar ‘double inductive’ structure to that possessed by the class of Reed-Muller (RM) codes. The new codes are structured such that their length-to-distance ratio n/d = 2m, m ≥ 2. For rates less than 1/2, the new codes have a higher rate than the next Reed-Muller code of the same order and length n = 2 · 2m.

Further results on Goppa codes and their applications to constructing efficient binary codes

It is shown that Goppa codes with Goppa polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{g(z)\}^{q}</tex> have the parameters: length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \leq q^{m} - s_{o}</tex> , number of check symbols <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n - k \leq m (q - 1) (\deg g)</tex> , and minimum distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d \geq q (\deg g) + 1</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> is a prime power, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is an integer, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(z )</tex> is an arbitrary polynomial over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(q^{m})</tex> , and so is the number of roots of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(z)</tex> which belong to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(q^{m})</tex> . It is also shown that all binary Goppa codes of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \leq 2^{m} - s_{o}</tex> satisfy the relation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n - k \leq m (d - 1)/2</tex> . A new class of binary codes with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \leq 2^{ m} + ms _{0}, n - k \leq m (\deg g) + s_{0}</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d \leq 2(\deg g) + 1</tex> is constructed, as well as another class of binary codes with slightly different parameters. Some of those codes are proved superior to the best codes previously known. Finally, a decoding algorithm is given for the codes constructed which uses Euclid's algorithm.

New classes of binary codes constructed on the basis of concatenated codes and product codes

We present new classes of binary codes that are constructed on the basis of concatenated codes and product codes. We discuss the random-error-correction capabilities of these codes. Some examples of the codes for the correction of random errors are given which have at least as many codewords as the best codes previously known (to the authors) with the same minimum distance and same number of check symbols. The burst-error-correction capabilities of the codes are also discussed. Several examples of the codes for the correction of both random errors and burst errors are given. A decoding algorithm for the codes is also described.

Sample-based classification procedures related to empiric distributions

We present new classes of binary codes that are constructed on the basis of concatenated codes and product codes. We discuss the random-error-correction capabilities of these codes. Some examples of the codes for the correction of random errors are given which have at least as many codewords as the best codes previously known (to the authors) with the same minimum distance and same number of check symbols. The burst-error-correction capabilities of the codes are also discussed. Several examples of the codes for the correction of both random errors and burst errors are given. A decoding algorithm for the codes is also described.

Some efficient binary codes constructed using Srivastava codes (Corresp.)

In this correspondence, we construct a new class of binary codes by exploiting the symmetry properties of the parity check matrix of the Srivastava codes. The construction is a generalization of Goppa's construction [1]. A number of the binary codes constructed are proved equal, or superior, to the best codes previously known.

A new class of binary codes constructed on the basis of BCH codes (Corresp.)

In this correspondence, we present a new class of binary codes that are constructed on the basis of BCH codes. Some examples of these codes are given, having more codewords than the best codes previously known (to the authors) with the same minimum distance and number of check symbols. A decoding algorithm for the codes is also described.

Some results on the problem of constructing asymptotically good error-correcting codes

Justesen has shown that concatenating a class of binary codes with a Reed-Solomon (RS) code produces asymptotically good codes. For low rates, the value of the ratio of minimum distance to code length (d/n) for such codes is substantially lower than that known to be achievable by the Zyablov bound. In this paper, we present a small class of binary codes with some useful properties. This class is then used in Justesen's construction to produce codes that have relatively large values of d/n for low rates.

Class of binary codes

Class of binary codes

Nearly perfect binary codes

A class of binary codes, satisfying the equality in a specialized version of the Johnson bound, is introduced, which contains perfect codes and the Preparata 2-error correcting codes. These codes are shown to contain t-designs, which can be extended to ( t + 1)-designs. It is shown how the weight-distribution of these codes can be uniquely derived.

The equivalence of rank permutation codes to a new class of binary codes (Corresp.)

An equivalence between the rank permutation codes and a new class of binary codes has been observed. A binary code may be generated by direct transformation of a permutation code. The binary codes are usually nonlinear and may be decoded by the inverse transformation and rank correlation of the equivalent permutation.

Combinatorial lemma and structure of a class of binary codes

A class of complete binary code systems is presented in which one of the symbols of the alphabet is used in any code combination not more than once, in terms of prefix codes of this class and of the operations of inversion and substitution of codes.

Coding theorems for the nonsynchronized channel

Capacity and error bounds are derived for a memoryless binary symmetric channel with the receiver having no a priori information as to the starting time of the code words. The channel capacity is the same as the capacity of the synchronized channel. For all rates below capacity, the minimum probability of error for the nonsynchronized channel decreases exponentially with the code-block length. For rates near channel capacity, the exponent in the upper bound on the probability of error for the nonsynchronized channel is the same as the corresponding exponent for the synchronized channel. For low rates, the largest exponent obtained for the nonsynchronized channel with conventional block coding is inferior to the exponent obtained for the synchronized channel. Stronger results are obtained for a new form of coding that allows for a Markov dependency between successive code words. Bounds on the minimum probability of error are obtained for unconstrained binary codes and for several classes of parity-check codes and are used to obtain asymptotic distance properties for various classes of binary codes. At certain rates there exist codes whose minimum distance, in the comma-free sense, is not only greater than one, but is proportional to the block length.

Encoding and error-correction procedures for the Bose-Chaudhuri codes

Bose and Ray-Chaudhuri have recently described a class of binary codes which for arbitrary m and t are t -error correcting and have length 2^m - 1 of which no more than mt digits are redundancy. This paper describes a simple error-correction procedure for these codes. Their cyclic structure is demonstrated and methods of exploiting it to implement the coding and correction procedure using shift registers are outlined. Closer bounds on the number of redundancy digits are derived.