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Analysis of Blind Reconstruction of BCH Codes

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Analysis of Blind Reconstruction of BCH Codes entropy Article Analysis of Blind Reconstruction of BCH Codes Soonhee Kwon and Dong-Joon Shin * Department of Electronics and Computer Engineering, Hanyang University, Seoul 04763, Korea; [email protected] * Correspondence: [email protected] Received: 16 October 2020; Accepted: 4 November 2020; Published: 5 November 2020 Abstract: In this paper, the theoretical lower-bound on the success probability of blind reconstruction of Bose–Chaudhuri–Hocquenghem (BCH) codes is derived. In particular, the blind reconstruction method of BCH codes based on the consecutive roots of generator polynomials is mainly analyzed because this method shows the best blind reconstruction performance. In order to derive a performance lower-bound, the theoretical analysis of BCH codes on the aspects of blind reconstruction is performed. Furthermore, the analysis results can be applied not only to the binary BCH codes but also to the non-binary BCH codes including Reed–Solomon (RS) codes. By comparing the derived lower-bound with the simulation results, it is confirmed that the success probability of the blind reconstruction of BCH codes based on the consecutive roots of generator polynomials is well bounded by the proposed lower-bound. Keywords: blind reconstruction; BCH codes; galois field; galois field fourier transform; lower-bound; RS codes 1. Introduction In order to achieve reliable information transmission through noisy communication channels, the use of error-correcting codes (ECCs) in data-stream is indispensable [1]. By sharing the parameters of ECCs between the transmitter and the receiver, the errors occurred by communication channels can be detected or corrected at the receiver in a cooperative way. However, in a non-cooperative context, it is necessary to decode received (or intercepted) data without the knowledge of parameters of the used ECC. In other words, a blind reconstruction of the parameters of the used ECC should be performed by the receiver. A blind reconstruction of ECCs has been studied in various ways [2–19]. The blind reconstruction schemes of linear block codes are studied in [2–9], the blind reconstruction schemes of Bose–Chaudhuri–Hocquenghem (BCH) codes are studied in [10–15], and the blind reconstruction schemes of convolutional codes are studied in [16–19]. Most of the blind reconstruction schemes of ECCs take the dual code approach to reconstruct the dual code space of the used code by using the received codewords. Valembois [2] proposed a detection and recognition algorithm for binary linear codes by using the dual code property and Cluzeau [3] proposed a blind reconstruction method based on iterative decoding techniques by using the dual code property. Moreover, most of the blind detection methods of BCH codes are also based on the dual code approach. By using the properties of BCH codes, their parity-check matrices can be constructed through applying Galois field Fourier transform (GFFT) on the received codewords and many of the blind reconstruction methods of BCH codes are based on GFFT [10–15]. In the same manner, most of the blind reconstruction methods of convolutional codes are also based on the dual code approach [16–19]. To reconstruct the generator polynomial or generator matrix of convolutional code, its dual code is recognized preferentially. An analysis of the blind reconstruction of cyclic codes over binary erasure channel (BEC) is performed in [20]. Note that for BEC, the number and the locations of error bits in the received Entropy 2020, 22, 1256; doi:10.3390/e22111256 www.mdpi.com/journal/entropy Entropy 2020, 22, 1256 2 of 16 data-stream are known to the receiver. By using this property of BEC, a blind reconstruction scheme of binary cyclic codes is proposed and a lower-bound on the detection probability of this scheme is analyzed in [20]. However, many blind reconstruction schemes consider the binary symmetric channel (BSC) where the number and the locations of error bits in the received data-stream are not unavailable. Therefore, the analysis in [20] is not directly applicable to the blind reconstruction schemes considering the BSC. In this paper, the blind reconstruction of BCH codes over q-ary symmetric channel is mainly considered because BCH codes are a most widely used class of cyclic codes, especially in communication and storage systems and q-ary symmetric channel is a general form of BSC. Especially, the method in [15] shows the best blind reconstruction performance among the existing blind reconstruction methods of BCH codes, but the theoretical analysis of this method has not been performed yet. Therefore, by analyzing the properties of BCH codes on the aspects of blind reconstruction, a lower-bound on the success probability of the blind reconstruction method in [15] is derived. More specifically, the distribution of GFFT values of the received codewords is analyzed and the blind reconstruction method is formulated by using the conjugacy classes. By comparing the derived lower-bound with the simulation results, it is confirmed that the success probability of the blind reconstruction is well lower-bounded. Furthermore, the analysis of BCH codes on the aspects of blind reconstruction may lay a foundation for an analysis of other blind reconstruction methods of BCH codes based on GFFT. In Section2, definitions and properties of BCH codes and GFFT are briefly explained. In Section3, the theoretic analysis of the properties of BCH codes on the aspects of blind reconstruction is performed. In Section4, the blind reconstruction method in [ 15] is explained, and a lower-bound on the success probability of this blind reconstruction method is derived. The simulation results confirm that the success probability of the blind reconstruction method is well-bounded by the derived lower-bound. In Section5, conclusions are provided. 2. BCH Codes and Galois Field Fourier Transform In this section, the BCH codes and the Galois field Fourier transform (GFFT) are briefly described. 2.1. BCH Codes BCH codes is a class of linear block codes for forward error correction. Let GF(q) denote the Galois field (or finite field) of q elements and let BCHq(n, k) denote the BCH code with length n and dimension k over GF(q). Note that the dimension k is the same as the length of random message which also implies the number of codewords. Then, the generator polynomial of BCHq(n, k) is defined as follows: g(x) = LCM[Mab (x), Mab+1 (x), ··· , Mab+d−2 (x)] (1) where LCM denotes the least common multiple function, a is a primitive n-th root of unity in GF(qm), i Mai (x) is a minimal polynomial of a over GF(q), b is an arbitrary positive integer smaller than n, and d is a designed distance. Note that m is the smallest integer such that n divides qm − 1. By the definition of generator polynomial g(x), ab, ab+1, ··· , ab+d−2 are the roots of g(x), i.e., g(ab) = g(ab+1) = ··· = b+d−2 g(a ) = 0. Let Sr be the set of the exponents of all roots of g(x) as follows: i Sr = i j g(a ) = 0, i 2 f0, 1, ··· , n − 1g . (2) k−1 A message can be expressed in polynomial form as m(x) = m0 + m1x + ··· + mk−1x and in vector form as m = (m0, m1, ··· , mk−1), where mi 2 GF(q) for i 2 f0, 1, ··· , k − 1g. A codeword of n−1 BCHq(n, k) can be expressed in polynomial form as c(x) = c0 + c1x + ··· + cn−1x and in vector form as c = (c0, c1, ··· , cn−1), where ci 2 GF(q) for i 2 f0, 1, ··· , n − 1g. Then, c(x) can be obtained as follows: c(x) = m(x)g(x). (3) Entropy 2020, 22, 1256 3 of 16 Since a codeword c(x) has g(x) as a factor, all roots of g(x) are also roots of c(x), i.e., c(ai) = 0 for all i 2 Sr. In this paper, the q-ary symmetric channel with error probability e is considered. Channel error n−1 can be expressed in polynomial form as e(x) = e0 + e1x + ··· + en−1x and in vector form as e = (e0, e1, ··· , en−1), where ei 2 GF(q) for i 2 f0, 1, ··· , n − 1g. Note that by the definition of q-ary symmetric channel, Pr(ei = 0) = 1 − e and Pr(ei = x) = e/(q − 1) for i 2 f0, 1, ··· , n − 1g and x 2 GF∗(q) where GF∗(q) = GF(q) n f0g. Then, a received codeword at the receiver is expressed in polynomial form as r(x) = c(x) + e(x), (4) or in vector form as follows: r = c + e. (5) Throughout the paper, the polynomial form and the vector form will be used interchangeably. i If there is no error (i.e., e(x) = 0), r(a ) = 0 for all i 2 Sr because r(x) = c(x). However, if e(x) 6= 0, i i we may have r(a ) 6= 0 for some i 2 Sr because it can be e(a ) 6= 0 for some i 2 Sr. 2.2. Conjugacy Classes and Cyclotomic Cosets m q q2 Let Ub denote a conjugacy class of b 2 GF(q ). Then, Ub consists of b and its conjugates b , b , 3 bq , ··· . Note that the conjugacy classes of the elements in the same conjugacy class are the same. i m The minimal polynomial of a 2 GF(q ) over GF(q), Mai (x), can be obtained by using the conjugacy classes as follows: Mai (x) = ∏ (x − z). (6) z2U ai The degree of Mai (x) is equal to jUai j, where jSj is the cardinality of a set S. Note that since Mai (x) has all the elements in Uai as its roots, g(x) in (1) has all the elements in Uab , Uab+1 , ··· , Uab+d−2 as its roots.
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