Information-theoretic interpretation of quantum error-correcting codes

Quantum error-correcting codes are analyzed from an information-theoretic perspective centered on quantum conditional and mutual entropies. This approach parallels the description of classical error correction in Shannon theory, while clarifying the differences between classical and quantum codes. More specifically, it is shown how quantum information theory accounts for the fact that "redundant" information can be distributed over quantum bits even though this does not violate the quantum "no-cloning" theorem. Such a remarkable feature, which has no counterpart for classical codes, is related to the property that the ternary mutual entropy vanishes for a tripartite system in a pure state. This information-theoretic description of quantum coding is used to derive the quantum analogue of the Singleton bound on the number of logical bits that can be preserved by a code of fixed length which can recover a given number of errors.Comment: 14 pages RevTeX, 8 Postscript figures. Added appendix. To appear in Phys. Rev.

Codes for Asymmetric Limited-Magnitude Errors With Application to Multilevel Flash Memories

Several physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions and bounds for such channels when the number of errors is bounded by t and the error magnitudes are bounded by ℓ. The constructions utilize known codes for symmetric errors, over small alphabets, to protect large-alphabet symbols from asymmetric limited-magnitude errors. The encoding and decoding of these codes are performed over the small alphabet whose size depends only on the maximum error magnitude and is independent of the alphabet size of the outer code. Moreover, the size of the codes is shown to exceed the sizes of known codes (for related error models), and asymptotic rate-optimality results are proved. Extensions of the construction are proposed to accommodate variations on the error model and to include systematic codes as a benefit to practical implementation

A Multi-Kernel Multi-Code Polar Decoder Architecture

Polar codes have received increasing attention in the past decade, and have been selected for the next generation of wireless communication standard. Most research on polar codes has focused on codes constructed from a $2\times2$ polarization matrix, called binary kernel: codes constructed from binary kernels have code lengths that are bound to powers of $2$. A few recent works have proposed construction methods based on multiple kernels of different dimensions, not only binary ones, allowing code lengths different from powers of $2$. In this work, we design and implement the first multi-kernel successive cancellation polar code decoder in literature. It can decode any code constructed with binary and ternary kernels: the architecture, sized for a maximum code length $N_{max}$, is fully flexible in terms of code length, code rate and kernel sequence. The decoder can achieve frequency of more than $1$ GHz in $65$ nm CMOS technology, and a throughput of $615$ Mb/s. The area occupation ranges between $0.11$ mm$^2$ for $N_{max}=256$ and $2.01$ mm$^2$ for $N_{max}=4096$. Implementation results show an unprecedented degree of flexibility: with $N_{max}=4096$, up to $55$ code lengths can be decoded with the same hardware, along with any kernel sequence and code rate