A New Zigzag-Decodable Code with Efficient Repair in Wireless Distributed Storage

A code is said to possess the combination property if <inline-formula><tex-math notation="LaTeX">$k$</tex-math> <alternatives><inline-graphic xlink:href="wang-ieq1-2591537.gif"/></alternatives></inline-formula> source packets are mapped into <inline-formula><tex-math notation="LaTeX">$n \geq k$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq2-2591537.gif"/></alternatives></inline-formula> packets and any <inline-formula> <tex-math notation="LaTeX">$k$</tex-math><alternatives><inline-graphic xlink:href="wang-ieq3-2591537.gif"/> </alternatives></inline-formula> out of these <inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq4-2591537.gif"/></alternatives></inline-formula> packets are able to recover the information of the original <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq5-2591537.gif"/></alternatives></inline-formula> packets. While the class of maximum-distance-separable codes are well known to have this property, its decoding complexity is generally high. For this reason, a new class of codes which can be decoded by the zigzag-decoding algorithm is considered. It has a lower decoding complexity at the expense of extra storage overhead in each parity packet. In this work, a new construction of a zigzag decodable code is proposed. The novelty of this new construction lies in the careful selection of the amount of bit-shift of each source packet in obtaining each parity packet. Besides, an efficient on-the-air repair scheme based on physical-layer network coding is designed.

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