We can see that the change between LlR 39 andLlR 40 is the largest, i. Discussion of the Element a Up to now, we have estimated the code roots from the set ja1,a2,a2'"—2 j but the element a2 —11 is ignored. To get high reliability, we choose the root that has the highest LLR.
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Then we re-rank the LLRs and estimate the code roots according to the previous steps. Re-execute the code roots recognition steps, it is easy to verify that the element binare fault codes —11 is not a root of the code and the case of Figure 1 does not appear in this algorithm.
Primitive Polynomial Recognition In Section 2 and Section 3, the primitive polynomial of the being recognized code is not considered.
As a dynamic binary instrumentation tool, instrumentation is performed at run time on the compiled binary files. Thus, it requires no recompiling of source code and can support instrumenting programs that dynamically generate code. Pin provides a rich API that abstracts away the underlying instruction-set idiosyncrasies and allows context information such as register contents to be passed to the injected code as parameters. Pin automatically saves and restores the registers that are overwritten by the injected code so the application continues to work.
But in fact, the corresponding primitive polynomial should be given when discussing an extension field GF 2mbecause there are more than one primitive polynomials in GF 2m and the calculation rules based on different primitive polynomials are not the same.
However, we propose a theorem that a binary cyclic codeword based on a primitive polynomial is also a valid binary cyclic codeword based on another primitive polynomial.
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According to this, we can choose any one primitive polynomial p x randomly and estimate the code length and code roots based on p x. Then, we can recognize the actual primitive polynomial according to the root properties of the BCH codes.
Theorem 1: A binary cyclic codeword Cr, which is encoded based on a primitive polynomial pj x over GF 2mis also a valid binary cyclic codeword based on another primitive polynomial p2 x over GF 2m with the same number of code roots.
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We define a and ft to be the roots of pj x and p2 xrespectively. According to Theorem 2.
Consequently, a codeword, which is based on a primitive polynomial pj xis also a valid codeword based on any other primitive polynomial p2 xand the number of code roots and error-correcting capabilities are the same. Therefore, when recognizing the parameters of a binary cyclic code, we can just randomly choose a primitive polynomial provisionally. In order to reduce the computational complexity, we recommend choosing the primitive polynomial with the smallest number of terms.
According to the basic character of the BCH codes, a generator polynomial has 2t roots with consecutive degrees the 2t roots do not form all the code roots, but include all the distinct rootswhere t is the correction capability of the codes.
In other words, ifa is a primitive element in GF 2mthe generator polynomial g x of a BCH code binare fault codes correcting t errors has a,a2,a3.
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Then, we can traverse all the primitive polynomials over GF 2m and get the one that makes the code roots be in accordant with the character of BCH codes as the primitive polynomial of the being recognized code. As an example, we still consider the BCH 63, 51 code referred in Section 3. The first six ones and the last six ones are two groups of conjugate roots. As a result, we can get the code roots p15, p30, p60, p57, p51, p39, p31, p62, p61, p59, p55, and p47 after the recognition as shown in Figure 3.
It is easy to verify that the estimated code roots also form two groups of conjugate roots, thus binare fault codes error-correction capability t equals two. Now we traverse all the other primitive polynomials and find the one under which the code roots are in accordant with the characters of BCH codes.
Figure 3. Recognizing the BCH 63, 51 codes using a primitive polynomial different from that of the encoder.
Computational Complexity In the proposed soft decision recognition algorithm, the most complex computation is the calculation ofpi. The major computational consumption appears in Equation 23which includes the neth root calculations, productions and tanh function in the real number field.
While, the hard decision algorithm only has production and addition calculations over GF 2. However, our proposed algorithm utilizes the soft decision outputs of the channel, which can provide more information about the reliability decision bits, so require very lower calculation times ofpjj than the hard decision one, which can also reduce the total computational complexity in the recognition procedure.
Simulations The simulation results of the proposed blind recognition algorithm are shown in this section.
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For each configuration, the information symbols in the codes are randomly chosen and the modulation mode is BPSK. When applying the algorithm to BCH 63,45 codes, the simulation results of the code length estimation are shown in Figures In Figure 4, we draw the stems of pt on different elements a' taken from GF 2m when the code length and synchronization positions are correct.
It is shown in Figure 4 that the values ofpt on the code roots are obviously higher than those on the other elements. And we also draw the stems on a fault code length and synchronization position in Figure 5.
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Figure 4. Values ofpt on correct code length and synchronization positions.
Binare fault codes result is in accordance with the simulation settings. Figure 5.
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Values ofpt on incorrect code length and synchronization positions. Figure 6. Code length and synchronization positions estimation of BCH 63, 45 codes. The performance of the algorithm is affected by the channel quality.
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In Figure 7, we draw the performance of the proposed algorithm when applied to code length and synchronization positions recognitions for several different binary BCH codes, binare fault codes the shortened codes. The curves depict the false recognition probabilities FRP of the code length and coding starting positions estimations on different SNRs. We also compare the performance of our proposed recognition algorithm with the hard-decision-based RIDERS algorithm proposed in .
After the code length and synchronization position estimation, the generator polynomial can be recognized by searching for the code roots according to the steps proposed in Section 3. The curves show the false recognition probabilities on different noise levels.
We also compare our proposed algorithm with the previous hard-decision-based recognition algorithms proposed by  in the figure.
It shows that the recognition performance is improved obviously in soft decision situations. A gap of dB exists between the two groups of curves.
Figure 7. How to make money by investing a dollar of code length estimations for some binary BCH codes. Performances of code roots estimations for some binary BCH codes. The code length estimation and block synchronization are achieved by checking the minimal parity check matrix.
After that, the code rate and generator polynomials are reconstructed by searching for the code roots. To the best of our knowledge, this paper is the first publication in literature which introduces an approach for complete-blind recognition of binary BCH codes in soft decision situations.
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Simulations show that our proposed blind recognition algorithm yields better performance than that of the previous hard-decision-based ones. Appendix: Proof of the faultiness of the Hypothesis 1 In this Appendix, we present that the Hypothesis 1 proposed in  is not always correct.
The proof is shown below: Proof. We let c x be the codeword polynomial of C.