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IEEE Transactions on Information Theory

Error Control Codes for Parallel Asymmetric Channels

Rudolf Ahlswede, Harout Aydinian

Published Jan 29, 2008• 6 citations
Publisher

Abstract

We consider transmission of messages encoded into binary matrices to be send over a parallel asymmetric channel. Two different channel models are introduced. They depend on the conditions on errors in the subchannels. For each channel model codes are constructed for error correction and for error detection and bounds are derived.
DOI
10.1109/tit.2007.913269

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Barrier-error channels have been suggested as a model for non-binary channels that are milder than symmetric-error channels. Such channels are motivated by practical applications in data storage and communications. The barrier-error model allows errors only to (downward) and from (upward) a specific symbol within the alphabet. We study a generalization of prior barrier-error models that considers different numbers of downward and upward errors. The results of this paper include a sufficient condition for error correction, code-size upper and lower bounds, a code construction decomposing to just two constituent symmetric-error codes (prior ones used many), and a decoding algorithm that decodes the two codes jointly. The decoder is shown empirically to achieve better block error rates when compared to previous algorithms.
Recently, the theory of quantum error control codes has been extended to subsystem codes over symmetric and asymmetric quantum channels -- qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error control codes has focused on code constructions for symmetric quantum channels. In this paper, we develop a theory and establish the connection between asymmetric quantum codes and subsystem codes. We present families of subsystem and asymmetric quantum codes derived, once again, from classical BCH and RS codes over finite fields. Particularly, we derive an interesting asymmetric and symmetric subsystem codes based on classical BCH codes with parameters [[n,k,r,d]]q[[n,k,r,d]]_q[[n,k,r,d]]q​, [[n,k,r,dz/dx]]q[[n,k,r,d_z/d_x]]_q[[n,k,r,dz​/dx​]]q​ and [[n,k′,0,dz/dx]]q[[n,k',0,d_z/d_x]]_q[[n,k′,0,dz​/dx​]]q​ for arbitrary values of code lengths and dimensions. We establish asymmetric Singleton and Hamming bounds on asymmetric quantum and subsystem code parameters; and derive optimal asymmetric MDS subsystem codes. Finally, our constructions are well explained by an illustrative example. This paper is written on the occasion of the 50th anniversary of the discovery of classical BCH codes and their quantum counterparts were derived nearly 10 years ago.
The binary case was studied in [1], but the method used there doesn't give the tight answer for nonbinary cases and we presented in [2] another method for the corresponding result.Here we formulate the main theorem and prove the auxiliary statements used in [2].During the transmission of q-ary words of length n over the channel at most t errors occur, and the encoder knows the set E of t positions, where these errors are possible.The decoder doesn't know anything about these positions.Let E t = E | E ⊆ {1, 2, . . ., n}, |E| = t be the set of all subsets from {1, 2, . . ., n} of size t and let M be a set of messages (|M| = M ) .A code word x(m, E) depends not only on the message m ∈ M but also on the configuration of possible errors E .So there exists the natural correspondence between the message m ∈ M and the list of code words E∈E t x(m, E) , which we use for the transmission of this message.Thus the code X for the set of messages M represents a collection of M lists E∈E t {x(m, E)}, m ∈ M .Sinde we can use the same word for different configurations, the size of a list can be essentially smaller than the size of the setLet us define the cylinder C(a, A) with the base a = (a, . . ., a n ) and the support A A ⊆ {1, 2, . . ., n} as the set of words (y 1 , . . ., y n ) with y i = a i , if i / ∈ A .It is clear that the size of the cylinder C(a, A) is equal to q |A| and the number of different cylinders with the same support A is equal to q n-|A| .As a result of the transmission of the codeword x(m, E) every word of C x(m, E), E can appear as output of the channel.The code X corrects t localized errors, if the decoder can correctly recover every message m ∈ M .The following condition is necessary and sufficient for it:
The theory of quantum error correcting codes is a primary tool for fighting decoherence and other quantum noise in quantum communication and quantum computation. Recently, the theory of quantum error correcting codes has developed rapidly and been extended to protect quantum information over asymmetric quantum channels, in which phase-shift and qubit-flip errors occur with different probabilities. In this paper, we generalize the construction of symmetric quantum codes via graphs (or matrices) to the asymmetric case, converting the construction of asymmetric quantum codes to finding matrices with some special properties. We also propose some asymmetric quantum Maximal Distance Separable (MDS) codes as examples constructed in this way.
We saw in Chapter 1 that the purpose of channel coding is to introduce redundancy to information messages so that errors that occur in the transmission can be detected or even corrected. In this chapter, we formalize and discuss the notions of error-detection and error-correction. We also introduce some well known decoding rules, i.e., methods that retrieve the original message sent by detecting and correcting the errors that have occurred in the transmission. Communication channels We begin with some basic definitions. Definition 2.1.1 Let A = { a 1 , a 2 , …, a q } be a set of size q , which we refer to as a code alphabet and whose elements are called code symbols . (i) A q-ary word of length n over A is a sequence w = w 1 w 2 … w n with each w i ∈ A for all i . Equivalently, w may also be regarded as the vector ( w 1 , …, w n ). (ii) A q-ary block code of length n over A is a nonempty set C of q -ary words having the same length n . (iii) An element of C is called a codeword in C . (iv) The number of codewords in C , denoted by ∣ C ∣, is called the size of C . (v) The (information) rate of a code C of length n is defined to be (log q ∣ C ∣)/ n . (vi) A code of length n and size M is called an (n, M)-code .