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Conference proceedings/Conference proceedings - IEEE International Conference on Systems, Man, and Cybernetics

Linear Programming Bounds for Multi-level Unequal Protection Codes

Tomohiko Saito, Toshiyasu Matsushima, Shigeichi Hirasawa

Published Oct 1, 2018• 1 citation
Publisher

Abstract

In coding theory, it is important to find upper bounds for the code size given a code length and minimum distance. The Hamming bounds and Linear Programming (LP) bounds were proposed in previous works. On the other hand, Masnick et al. proposed Unequal Error Protection (UEP) codes and modified Hamming bounds as upper bounds for the code size of UEP codes. In our previous work, we defined 2-level UEP codes as a subclass of UEP codes, and derived LP bounds for 2-level UEP codes. In this paper, we define multi-level UEP codes by extending 2-level UEP codes, and derive LP bounds for multi-level UEP codes. Moreover, we show that LP bounds for UEP codes are tighter upper bound than modified Hamming bounds.

DOI
10.1109/smc.2018.00496

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In coding theory, it is important to find upper bounds for the code size given a code length and minimum distance. The Hamming bounds and Linear Programming (LP) bounds were proposed in previous works. On the other hand, Masnick et al. proposed Unequal Error Protection (UEP) codes and modified Hamming bounds as upper bounds for the code size of UEP codes. In our previous work, we defined 2-level UEP codes as a subclass of UEP codes, and derived LP bounds for 2-level UEP codes. In this paper, we define multi-level UEP codes by extending 2-level UEP codes, and derive LP bounds for multi-level UEP codes. Moreover, we show that LP bounds for UEP codes are tighter upper bound than modified Hamming bounds.
In coding theory, it is important to find upper bounds for the code size given a code length and minimum distance. The Hamming bounds and Linear Programming (LP) bounds were proposed in previous works. On the other hand, Masnick et al. proposed Unequal Error Protection (UEP) codes and modified Hamming bounds as upper bounds for the code size of UEP codes. In our previous work, we defined 2-level UEP codes as a subclass of UEP codes, and derived LP bounds for 2-level UEP codes. In this paper, we define multi-level UEP codes by extending 2-level UEP codes, and derive LP bounds for multi-level UEP codes. Moreover, we show that LP bounds for UEP codes are tighter upper bound than modified Hamming bounds.
符号長n と最小距離d の誤り訂正符号に対し，符号語数の上界として，ハミング限界や線形計画(Linear Programming: LP) 限界が知られている．一方，Masnick らによって不均一誤り訂正(Unequal Error Protection: UEP) 符号が提案された．UEP 符号においても，符号語数の上界として，ハミング限界を拡張した修正ハミング限界が示されている．従来，著者らはUEP 符号のサブクラスとして2-レベルUEP符号を定義し，そのLP 限界を示した．本論文では，2-レベルUEP 符号を拡張した，多レベルUEP 符号を定義し，そのLP 限界を導出する．更に，多レベルUEP 符号のLP 限界が修正ハミング限界よりも優れていることを示す．
A lower bound on the length of binary self-orthogonal unequal error protection (UEP) codes is derived, and two design procedures for constructing optimal self-orthogonal UEP codes are proposed. With this lower bound, known self-orthogonal UEP codes can be evaluated. It is pointed out that, for given values of minimum distance and code rate, the self-orthogonal codes must be relatively long, so optimal self-orthogonal codes are not optimal in general. But self-orthogonal codes can be implemented simply, and they have error-correcting capabilities beyond those guaranteed by their minimum distance. These properties can be viewed as a partial compensation for using self-orthogonal codes.>
The decoding of unequal error protection product codes, which are a combination of linear unequal error protection (UEP) codes and product codes, is addressed. A nonconstructive proof of the existence of a good error-erasure-decoding algorithm is presented; however, obtaining the decoding procedure is still an open research problem. A particular subclass of UEP product codes is considered, including a decoding algorithm that is an extension of the Blokh-Zyablov decoding algorithm for product codes. For this particular subclass the decoding problem is solved.>
Irregular low-density parity check (LDPC) codes are particularly well-suited for transmission schemes that require unequal error protection (UEP) of the transmitted data due to the different connection degrees of its variable nodes. However, this UEP capability is strongly dependent on the connection profile among the protection classes. This paper applies a multi-edge type analysis of LDPC codes for optimizing such connection profile according to the performance requirements of each protection class. This allows the construction of UEP-LDPC codes where the difference between the performance of the protection classes can be adjusted and with an UEP capability that does not vanish as the number of decoding iterations grows.
In this paper, a new construction of quasi-cyclic low-density parity-check (QC-LDPC) codes for unequal error protection (UEP) is proposed. We first give a new class of UEP block codes. QC-LDPC codes with binomial-term parity-check matrices which can avoid girths less than 8 and achieve an enlarged upper bound on the minimum distance are also introduced. An effective UEP scheme based on the proposed UEP block codes and QC-LDPC codes which can provide flexible choice of protection levels is then presented. Simulation results show that the new QC-LDPC codes can achieve good performance and low error floors. The bits with different designed protection levels can indeed have unequal bit-error-rate performance.