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...ng turbo-codes to the quantum setting first requires a quantum analogue of convolutional codes. These have been introduced in [10], [11], [31], [32] and followed by further investigations [15], [18], =-=[1]-=-. Quantum turbo-codes can be obtained from the interleaved serial concatenation of convolutional codes. This idea was first introduced in [33]. There, it was shown that, on memoryless Pauli channels, ...

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...ion against noise and decoherence. A well-known class of quantum codes is called stabilize codes, in which it can be easily constructed using self-orthogonal (or dual-containing) classical codes, see =-=[3]-=-, [5], [11] and references therein. Recently, subsystem codes combine the features of decoherence free subspaces, noiseless subsystems, and quantum errorcorrecting codes, see [2], [4], [13], [14] and ...

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... the minimum distance of a BCH code is less than or equal to its designed distance. [15, 11, 3]2, [31, 26, 3]2 and [63, 56, 3]2 are examples of BCH codes that protect one and two link failures. Also, =-=[15, 8, 5]-=-2, [31, 21, 5]2 and [48, 36, 5]2 are examples of BCH codes against four link failures [15], [21]. In the next section we will include tables of the best known network protection codes. VII. CODE CONST...

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...onvolutional codes [3]. Recently, two of us constructed quantum convolutional codes from product codes [5] and derived an algorithm to construct non-catastrophic encoders and encoder inverses [6]. In =-=[1]-=-, a generalized Singleton bound for a class of quantum convolutional codes has been established, together with a family of codes based on generalized Reed-Solomon codes meeting this bound. Unit memory...

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...n self-orthogonal codes have been studied through algebraic and combinatorial tools [6], [7], [12]. In recent years, this topic has become increasingly interesting due to application to quantum codes =-=[1]-=-, [2], [5], [10], [11], [13]. For application to quantum codes, one is interested in not only classical Euclidean selforthogonal codes but also some other types of self-orthogonal codes such as Hermit...

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... and [[64, 9, {≥ 24, 2}]]4 were obtained using the concatenated RS code construction. 2.1 Quantum BCH Codes The idea of constructing CSS codes from BCH codes is not new, as Steane [22] and Aly et al. =-=[1, 2]-=- give quantum symmetric and asymmetric codes. The difficulty with their methods is the lack of knowledge of the dual distances, as noted in [19]. In addition, there is a rate gain which is linearly pr...

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...th n. A linear code defined by a generator polynomial g(x) has dimension k = n − deg(g(x)) and rate k/n. It was not an easy task to show the dimension of nonprimitive BCH codes over finite fields. In =-=[2]-=-, [3], we have given an explicit formula for the dimension of these codes if their deigned distance δ is less than a constant δmax. Low-density parity check (LDPC) codes are a capacityapproaching (Sha...