Channel decomposition for multilevel codes over multilevel and partial erasure channels

Carolyn Mayer; Kathryn Haymaker; Christine A. Kelley

doi:10.3934/amc.2018010

Article Contents

2018, Volume 12, Issue 1: 151-168. Doi: 10.3934/amc.2018010

This issue Previous Article On the spectrum for the genera of maximal curves over small fields Next Article On

${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$

-additive cyclic codes

Channel decomposition for multilevel codes over multilevel and partial erasure channels

1.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0130, USA
2.
Department of Mathematics & Statistics, Villanova University, Villanova, PA 19085, USA

* Corresponding author: Carolyn Mayer.
* Corresponding author: Carolyn Mayer.

Received: September 30, 2016

Published: February 2018

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Abstract

We introduce the Multilevel Erasure Channel (MEC) for binary extension field alphabets. The channel model is motivated by applications such as non-volatile multilevel read storage channels. Like the recently proposed $q$-ary partial erasure channel (QPEC), the MEC is designed to capture partial erasures. The partial erasures addressed by the MEC are determined by erasures at the bit level of the $q$-ary symbol representation. In this paper we derive the channel capacity of the MECand give a multistage decoding scheme on the MEC using binary codes. We also present a low complexity multistage $p$-ary decoding strategy for codes on the QPEC when $q = p^k$.We show that for appropriately chosen component codes, capacity on the MEC and QPEC may be achieved.

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Mathematics Subject Classification: Primary: 94A40, 94B35; Secondary: 94A15.

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Figure 1. $4$-ary partial erasure channel (QPEC) with $M=2$. Here, the binary representation of each $4$-ary symbol is shown.

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Figure 2. $4$-ary multilevel erasure channel with erasure probability $\varepsilon$ and bit error probability $\gamma$.

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Figure 3. $I(X;Y)$ as a function of $\varepsilon$, given a uniform input distribution.

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Figure 4. Subchannel for $X_1$ on $4$-ary multilevel erasure channel with parameters $\varepsilon,\gamma$.

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Figure 5. Subchannel for $X_2$ on $4$-ary multilevel erasure channel with parameters $\varepsilon,\gamma$.

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Figure 6. Subchannel for $X_1$ on 4-ary partial erasure channel (QPEC) with erasure probability $\varepsilon$.

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Figure 7. Subchannel for $X_2$ on 4-ary partial erasure channel (QPEC) with erasure probability $\varepsilon$.

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