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We propose efficient coding schemes for two communication settings: 1) asymmetric channels and 2) channels with an informed encoder. These settings are important in non-volatile memories, as well as optical and broadcast communication. The schemes are based on non-linear polar codes, and they build on and improve recent work on these settings. In asymmetric channels, we tackle the exponential storage requirement of previously known schemes that resulted from the use of large Boolean functions. We propose an improved scheme that achieves the capacity of asymmetric channels with polynomial computational complexity and storage requirement. The proposed non-linear scheme is then generalized to the setting of channel coding with an informed encoder using a multicoding technique. We consider specific instances of the scheme for flash memories that incorporate error-correction capabilities together with rewriting. Since the considered codes are non-linear, they eliminate the requirement of previously known schemes (called polar write-once-memory codes) for shared randomness between the encoder and the decoder. Finally, we mention that the multicoding scheme is also useful for broadcast communication in Marton’s region, improving upon previous schemes for this setting. [ABSTRACT FROM PUBLISHER]

The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. In this paper, we explore $[n,k,d]$ systematic error-correcting codes for rank modulation. Such codes have length $n$ , $k$ information symbols, and minimum distance $d$ . Systematic codes have the benefits of enabling efficient information retrieval in conjunction with memory-scrubbing schemes. We study systematic codes for rank modulation under Kendall’s $\tau $ -metric as well as under the $\ell _\infty $ -metric. In Kendall’s $\tau $ -metric, we present $[k+2,k,3]$ systematic codes for correcting a single error, which have optimal rates, unless systematic perfect codes exist. We also study the design of multierror-correcting codes, and provide a construction of $[k+t+1,k,2t+1]$ systematic codes, for large-enough $k$ . We use nonconstructive arguments to show that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes. Finally, in the $\ell _\infty $ -metric, we construct two $[n,k,d]$ systematic multierror-correcting codes, the first for the case of $d=O(1)$ and the second for $d=\Theta (n)$ . In the latter case, the codes have the same asymptotic rate as the best codes currently known in this metric. [ABSTRACT FROM PUBLISHER]