This paper considers polar coding for asymmetric settings, that is, channel coding for asymmetric channels and lossy source coding for nonuniform sources and/or asymmetric distortion measures. The difficulty for asymmetric settings comes from the fact that the optimal symbol distributions of codewords are not always uniform. It is known that such nonuniform distributions can be realized by Gallager's scheme which maps multiple auxiliary symbols distributed uniformly to an actual symbol. However, the complexity of Gallager's scheme increases considerably for the case that the optimal distribution cannot be approximated by simple rational numbers. To overcome this problem for the asymmetric settings, a new polar coding scheme is proposed, which can attain the channel capacity without any alphabet extension by invoking results on polar coding for lossless compression. It is also shown that the proposed scheme achieves a better tradeoff between complexity and decoding error probability in many cases.

Private information retrieval scheme for coded data storage is considered in this paper. We focus on the case where the size of each data record is large and hence only the download cost (but not the upload cost for transmitting retrieval queries) is of interest. We prove that the tradeoff between storage cost and retrieval/download cost depends on the number of data records in the system. We also propose a fairly general class of linear storage codes and retrieval schemes and derive conditions under which our retrieval schemes are error-free and private. Tradeoffs between the storage cost and retrieval costs are also obtained. Finally, we consider special cases when the underlying storage code is based on an MDS code. Using our proposed method, we show that a randomly generated retrieval scheme is indeed very likely to be private and error-free.

In the (k, n) threshold scheme, the information X is partitioned and coded into subinformation. If any k subinformation is obtained among n subinformation, the original information X can be recovered completely. However, no information can be obtained at all concerning X from any (k – 1) subinformation. Thus, the (k, n) threshold scheme is suited to the distributed storage or transmission of information. On the other hand, each subinformation requires the same number of bits as the original information X, which is very inefficient from the viewpoint of the coding efficiency. This paper extends the (k, n) threshold scheme and proposes the (k, L, n) threshold scheme. In the proposed scheme, the original information can be recovered completely from any k subinformation, but no information concerning X is obtained at all from any (k – L) subinformation. From any (k – t) subinformation (1 ≤ t ≤ L – 1), the information obtained for X contains the ambiguity of (t/L) H(X). In (k, L, n) scheme, the bit‐length of each subinformation is 1/L of the information X, which is a coding with very high efficiency. This paper presents a construction method for (k, L, n) threshold scheme, together with the discussion of its characteristics.

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