In this paper, we develop a new channel model, which we name the $q$-arypartial erasure channel (QPEC). The QPEC has a $q$-ary input, and its output iseither the input symbol or a set of $M$ ($2 \le M \le q$) symbols, containingthe input symbol. This channel serves as a generalization to the binary erasurechannel, and mimics situations when a symbol output from the channel is knownonly partially, that is, the output symbol contains some ambiguity, but is notfully erased. This type of channel is motivated by non-volatile memorymulti-level read channels. In such channels the readout is obtained by asequence of current/voltage measurements, which may terminate with partialknowledge of the stored level. Our investigation is concentrated on theperformance of low-density parity-check (LDPC) codes when used over thischannel, thanks to their low decoding complexity using belief propagation. Weprovide the exact QPEC density-evolution equations that govern the decodingprocess, and suggest a cardinality-based approximation as a proxy. We thenprovide several bounds and approximations on the proxy density evolutions, andverify their tightness through numerical experiments. Finally, we provide toolsfor the practical design of LDPC codes for use over the QPEC.
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机译：在本文中，我们开发了一个新的渠道模型，我们将其命名为$ q $ -partial擦除渠道（QPEC）。 QPEC具有$ q $ ary输入，其输出是输入符号或包含输入符号的一组$ M $（$ 2 \ le M \ le q $）符号。该通道用作二进制擦除通道的通用化，并模拟从通道输出的符号仅部分已知（即，输出符号包含一些歧义但被明显擦除）时的情况。这种类型的通道是由非易失性存储器多级读取通道驱动的。在这样的通道中，通过电流/电压测量的顺序来获得读数，这可能会随着对存储电平的部分了解而终止。我们的研究集中在低密度奇偶校验（LDPC）码在此通道上使用时的性能，这归因于它们使用置信传播的低解码复杂度。我们提供了精确的QPEC密度演化方程来控制解码过程，并提出了基于基数的近似作为代理。然后，我们提供了代理密度演化的几个边界和近似值，并通过数值实验验证了它们的紧密性。最后，我们提供了用于在QPEC上使用的LDPC代码的实际设计工具。
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