低译码错误概率的线性码的码率
Rates of linear codes with low decoding error probability
摘要：
考虑如下从二部图获得的二进制线性码。有~ \(k \geq 1\)左节点，每个代表一个消息位，有~ \(m = m(k)\)右节点，每个代表一个奇偶校验位，从相应的消息节点邻居集合中产生。消息和奇偶校验位都通过无记忆二进制输入通道发送，该通道独立地保留、翻转或擦除每个传输位。基于接收到的符号集，接收机处的解码器获得对所发送的原始消息的估计。如果解码错误概率~ \(P_k \longrightarrow 0\)和每个奇偶节点的平均度保持有界为~ \(k \rightarrow \infty,\)，则码的速率~ \(\frac{k}{k+m} \longrightarrow 0\)为~ \(k \rightarrow \infty.\)
英文摘要：
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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