Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rates of linear codes with low decoding error probability   Consider binary linear codes obtained from bipartite graphs as follows. There
are~\(k \geq 1\) left nodes each representing a message bit and there are~\(m =
m(k)\) right nodes each representing a parity bit, generated from the
corresponding set of message node neighbours. Both the message and the parity
bits are sent through a memoryless binary input channel that either retains,
flips or erases each transmitted bit, independently. Based on the received set
of symbols, the decoder at the receiver obtains an estimate of the original
message sent. If the decoding error probability~\(P_k \longrightarrow 0\) and
the average degree per parity node remains bounded as~\(k \rightarrow \infty,\)
then the rate of the code~\(\frac{k}{k+m} \longrightarrow 0\) as~\(k
\rightarrow \infty.\)

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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex