The present paper explains how this problem has been resolved even for additive white Gaussian channel by tracing the long history of classical and quantum information theory.Ĭurrently, finite block-length theory is one of hottest topics in information theory and is discussed more precisely for various situations elsewhere. Therefore, the successive researchers had to recover his result without use of his derivation. Also, in spite of the importance of his result, many researchers overlooked his result because his paper was written in German. Although he derived the second-order coefficient for the discrete channel, he could not derive it for the additive white Gaussian channel. The calculation of the second-order coefficient approximately gives the solution of the above problem, that is, the real optimal transmission rate with finite block-length n.
On the other hand, in 1962, Strassen 5) addressed this problem by discussing the coefficient with the order of the transmission rate, which is called the second-order asymptotic theory.
#Tajima pulse attach security device code
Here, we should emphasize that any actually constructed code has a finite block-length and will not necessarily attain the conventional asymptotic transmission rate. Hence, many researchers have doubted what the real optimal transmission rate is. However, still no actually constructed code could attain the optimal transmission rate. 3, 4) It was theoretically shown that they can attain the optimal transmission rate when the block-length n goes to infinity. Since the 1990s, turbo codes and low-density parity check (LDPC) codes have been actively studied as useful codes. However, although these codes realize a sufficiently small error probability, no code could attain the optimal transmission rate. Many practical codes have been proposed, depending on the strength of the noise in the channel, and have been used in real communication systems. To construct a practical code, we need another type of theory, which is often called coding theory. However, we cannot directly apply the channel coding theorem to actual information transmission because this theorem guarantees only the existence of a code with the above ideal performance. In the case of an additive white Gaussian channel, the channel coding theorem is that the optimal transmission rate is, where is the signal-noise ratio. In the following, instead of this term, we employ the transmission rate, which expresses the number of transmitted bits per one use of the channel, to characterize the speed of the transmission. In this case, we cannot use the term redundancy because its meaning is not clear. We can consider a similar problem when the channel is given as additive white Gaussian noise. Under these conditions, the limit of the minimum error probability depends only on whether the rate of the redundancy is larger than the entropy H( P) or not. This fact is called the channel coding theorem. He showed that we can recover the original message by a suitable code when the noise of each bit is independently generated subject to the probability distribution P, the rate of redundancy is the entropy H( P), and the block-length n is infinitely large. To discuss this problem, for a probability distribution P, he introduced the quantity H( P), which is called the (Shannon) entropy and expresses the uncertainty of the probability distribution P. In particular, he clarified the minimum redundancy required to correct an error with probability almost 1 with an infinitely large block-length n. In 1948, Shannon 1) discovered that increasing the block-length n can improve the redundancy and the range of correctable errors. The reason for the large redundancy in the simple code described above is that the block-length (the number of bits in one block) of the code is only 3. For practical use, we need to improve on this code, that is, decrease the amount of redundancy and enlarge the range of correctable errors. For example, if two bits are flipped during the transmission, we cannot recover the original message. In this example, the code has a large redundancy and the range of correctable errors is limited.
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