An Efficient Decoding Algorithm for Block Codes Based on the Communication Channel Reliability Information

For channel codes in communication systems, an efficient algorithm that controls error is proposed. It is an algorithm for soft decision decoding of block codes. The sufficient conditions to obtain the optimum decoding are deduced so that the efficient method which explores candidate code words can be presented. The information vector of signal space codes has isomorphic coherence. The path metric in the coded demodulator is the selected components of scaled regions. The carrier decision is derived by the normalized metric of synchronized space. An efficient algorithm is proposed based on the method. The algorithm finds out a group of candidate code words, in which the most likely one is chosen as a decoding result. The algorithm reduces the complexity, which is the number of candidate code words. It also increases the probability that the correct code word is included in the candidate code words. It is shown that both the error probability and the complexity are reduced. The positions of the first hard-decision decoded errors and the positions of the unreliable bits are carefully examined. From this examination, the candidate codewords are efficiently searched for. The aim of this paper is to reduce the required number of hard-decision decoding and to lower the block error probability.

To determine c , we should search for a few candidate codewords.

2-1-Sufficient Condition
Let the first candidate code word be 1 11 12 1 ( , , , ) n c c c  c , which is obtained by hard-decision decoding of y . However, uncorrectable but detectable errors may occur, for some codes which is not perfect. If uncorrectable errors are detected, the least confident bit of y is complemented (i.e., if i a is the smallest, complement i y ) and a hard-decision decoding is performed again to obtain 1 c . But, if uncorrectable errors are detected again, we finish the entire decoding procedure reporting the detection of errors. The path metric represents the quantized components in the demodulator [6]. The optimum decision is based on normalized metric of the synchronized space [8]. The scaled regions are designed to achieve the uniform phase [7,8]. The symmetry properties of signal space codes are isomorphic to the signal set in the demodulator [9]. The carrier decision is derived by the normalized metric of synchronized space [4,6]. The adjacent measure has constant weight with the symbol code which gives zeros in the synchronized space. The maximum likelihood unrestricted decision of redundant bits determines the decision threshold [4].
The code point that is nearest to the decoding result generates the candidate code words [10]. The subsets of the prefix length matching parameters provide optimum gain. The coding gain is obtained by symmetric search. The carrier phase should be considered. When the PLL synchronizer is used, the symbol parameters are efficiently estimated. If we can obtain 1 c , the error pattern for 1 c is given by 1

2-2-Locating the Error Positions
Starting from a code word c and its error pattern e , we explore other candidate code words. Let be a vector whose Hamming weight is integer w . We set The vector w b may not be a code word. We hard decision decode w b to obtain a code word w c . We obtain a candidate code word  One can properly select scaled transformation to be applied to these lattices in order to produce the required number of representatives in which () a W  eb will be partitioned. Let ( , ) xy be a prime solution to ( , ) A x y q  for any integer q . Let ( , ) A x y be defined as the sum of two squares, that is,

( , )
A x y x y . The solution is connected to the problem of realizing the partitioning of a lattice into sublattices. This interest is basically due to the richness of the algebraic structure available and the possibilities to explore new ways of constructing complex lattices as well as of partitioning them. However, we also need a method for constructing irreducible decomposable forms in order not to be too restrictive.
In this direction, let  be any algebraic number field of degree n , and let  be a primitive element for The minimum monic polynomial () px of the number  over the field Q has degree n . An extension L over  can be constructed for () px factoring completely, The coefficients of the field are rational numbers such that the norm () x be elements of Q , then the norm of the number 1 1 2 2 Is irreducible over Q . The integral solutions to Let the set M D be the coefficients ring of the module M . Since the element l is in M D , M D is a ring with unit. The ring of integers of the field  is a full module in the field of algebraic number which contains the number l and is a ring. Consequently, the ring of the algebraic number field Q is the ring of integers of the field. Recalling the problem of integral representation by decomposable forms, we see that it reduces to the determination in a full module of all numbers  for which () Nq   . Now, for any element of D such that () Nl   , the product   is in M . Thus, the coefficients provide new solutions of () Nq   from one solution.
Page | 339 The coefficients belong to the set of elements which are the ring D . If  is an element of  , the units of D satisfy the sufficient condition. Consequently, if  is an element of  , the polynomial has integer coefficients. Since there are various orders of a number field  , there is a maximal order containing the previous ones because there are various orders of a number field  .
There is a geometric representation of the algebraic numbers as points in n -dimensional space. Let  be a number field. The set { 12 Being a point of the space , st L . The binary quadratic forms , with a c l  and 0 b  ; and 1 abc    , are irreducible rational forms. Which decomposes into linear factors in some quadratic field (extension of the rational field of degree 2). Therefore, the full modules in quadratic fields is the connection with the problem of representation of integers by the binary quadratic forms. Being more specific, let  be a field, ( Evaluating (9) If some classes of modules have coefficient ring, the corresponding forms also have the same discriminant.

3-Decoding Algorithm
The information vector of signal space codes has isomorphic coherence. The path metric in the coded demodulator is the selected components of scaled regions. The optimum decision is based on normalized metric of synchronized space. The scaled regions are designed to achieve the uniform phase.
The positions of the first hard-decision decoded errors and the positions of the unreliable bits are carefully examined. From this examination, the candidate codewords are efficiently search for. The aim of the algorithm is to reduce the required number of hard-decision decoding and to lower the block error probability.

(
Step 1) Decode y to obtain c with hard decision information.

(
Step 2) If the sufficient condition of optimum decoding for c is satisfied, then c is the code word. Terminate decoding.

4-Simulation Results
The algorithm is applied to some widely used block codes. The algorithm is compared with separate path equalization (SPA) algorithm [7]. Figures 2 and 3 show the decoding error probability. The signal-to-noise ratio is expressed as N are the energy per information bit and the single-sided noise spectral density, respectively [6]. Compared to the SPA algorithm, the proposed algorithm can increase the probability that the correct code word is included in the candidate words. The adaptive permutation of the proposed algorithm works efficiently. The proposed algorithm can reduce the decoding error probability. The proposed algorithm involves obtaining the most and least reliable bits by sorting the reliability vectors in ascending order. The normalized metric transformation stage of the algorithm takes a large component of the decoding complexity. The smallest analog weight requires more iterative steps, while the decoding procedure quickly arrives at the negative element stopping condition before the algorithm properly decodes the received vector. However, it is clear that the candidate error pattern is computationally efficient, since it yields a similar decoding performance with the smallest analog weight value and requires moderate number of iterations. The result in Figure 3 indicates that the proposed algorithm outperforms the SPA algorithm for the same input data. This is because the binary cyclic code of length 64 is designed to select scaled transformation. It can be applied to the lattices in order to produce the required number of representatives. The ring of integers of the field is a full module in the field of algebraic number which contains the number l and is a ring. The algorithm is suboptimal, but robust for decoding any class of linear block code and can be realized on a real-time coding scheme. Figures 4 and 5 show that the proposed algorithm reduces the complexity of decoding [2]. The decoding complexity means the number of iterations of hard-decision decoding. A soft-decision decoding includes multiple hard decision decoding processes. When we perform a soft-decision decoding, the most computational process is hard-decision decoding. Therefore, it is common to indicate the complexity of soft-decision decoding as the number of iterations of hard-decision decoding [6]. In the searching process, if the analog weight of the error pattern is less than a predicted threshold value, then we can safely skip the search for remaining candidate code words. As can be observed from Figures 4 and 5, the computational complexity of the proposed algorithm is low. When / bo EN = 4.0dB, the number of iterations of hard-decision decoding is only 3.846 for (23,12) Golay code, which is considerably small. In addition, as can be gleaned from the figures, the number of iterations of hard-decision decoding, which requires the highest computational complexity, decreases steeply as the signal-to-noise ratio increases. Those of the other orders also decrease as the signal-to-noise ratio increases, making the proposed algorithm very fast. The algorithm increases the probability that the correct code word is included in the candidate words. The algorithm reduces the number of harddecision decoding.
It has been shown that both the error probability and the complexity are reduced. The effect of exploring candidate code words with the proposed algorithm has been confirmed.

5-Conclusion
For block codes in communication systems, an efficient algorithm for error controlling is proposed. It is an algorithm for soft decision decoding of block codes. The aim of the algorithm is to reduce the required number of hard-decision decoding and to lower the block error probability. The algorithm finds out a group of candidate code words, in which the most likely one is chosen as a decoding result. It is an algorithm for soft decision decoding of block codes. The sufficient conditions to obtain the optimum decoding are deduced so that the efficient method which explores candidate code words can be presented. The information vector of signal space codes has isomorphic coherence. The path metric in the coded demodulator is the selected components of scaled regions. The carrier decision is derived by the normalized metric of synchronized space.
The transmitted word is included in the set of explored candidate code words. The classes of modules have coefficient ring. The corresponding forms also have the same discriminant modules in quadratic fields. The representation of integers by binary quadratic forms reduces to the problem of similarity of modules in quadratic fields. The position tracking is achieved by the concepts of the theory algebraic numbers such as the ring of integers, integral basis, and decomposition of forms in finite extension field. The required number of hard-decision decoding and the complexity of the algorithm are reduced. The number of candidate code words is reduced, too. The proposed algorithm increases the probability that the correct code word is included in the candidate words. It is shown that both the error probability and the complexity are reduced. The positions of the first hard-decision decoded errors and the positions of the unreliable bits are carefully examined. From this examination, the candidate codewords are efficiently searched for. An efficient and systematic way of soft decision decoding of block codes is provided.

6-Conflict of Interest
The author declares that there is no conflict of interests regarding the publication of this manuscript. In addition, the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancies have been completely observed by the authors.