Question 27.
Why did you write that you solved the problem of Shannon, if the distance to a Gaussian channel capacityfor the usage soft and hard modems, as it is clear from your last publications is still about 1 dB?
The answer 27.
You are absolutely right. And for the quantization of the AWGN channel input binary signal into 16 levels (4 bits), and in the case of hard modem that accepts only decisions "0" or "1" about the values of the received symbols when the channel turns into a simple BSC, this value is the distance to the energy level of the Shannon's bound for code rate R~1/2 and now for our MTD decoders it is about 1 dB. Perhaps if we announced the solution the problem of Shannon, when the distance would be more than 4 dB, then it would be toomuch. This would mean that the real transmission power by more than 2.5 times higher than the level for the channel capacity. This, of course, would be very weakrostrum to declare some sort of the scientific "victory". But in our case, 1 dB is exceeding the Shannon's bound only at 26%, i.e. just a quarter of its minimum possible level! It's a huge difference! And note that we, until recently, very quickly moved to this bound. In our last works we have already discussed, what are the opportunities for greater promotion the working region of decoder to the capacity. This fundamental work will continue.
But let us now remember that the discussed bound is absolutely unattainable. It becomes infinitely elastic as the movementto it. And unimportant difference of energy of 26% or perhaps 15% more to the level of this boundary in the development of those or other systems of communication will not change significantly the characteristics of such systems. Although, for example, for the deep Space communication additional half decibel energy can be quite useful.
Thus, our announcement about the successful solution the problem of Shannonis taken in the right time of achievement the significant level of proximity to the bound. Moreover, the energy level of MTD at R~1/2 is no more available for any other algorithms with reasonable complexity decoding in Gaussian channels, which is also very significant and very revealing.
Finally, the level of efficiency of MTD algorithms in nonbinary channels and in channels with erasures nowalso became so close to the maximum possible noise level that the value of the MTD results in all channels with independent distortion really allowed us to assert the solution of this great problem appealed 70 years ago. Moreover, MTD algorithms saved theoretically minimal complexity of decoding , linearly growing with the code length. This emphasizes the high scientific and practical significant achievements of our optimization theory (OT) coding theory, which is now developed at the scientific basis of searching global extremum of functional, of course, in the specific conditions of discrete mathematical spaces.
All currently known estimates for the number of decoding iterations in the MTD algorithms show that the number of iterations of the controlled symbol correction is very limited, and their real value in most cases does not exceed I=200. This is much less than it is needed for many classic iterative methods, which further emphasizes the advantage of MTD for implementation in all digital systems.
Question 28.
Why coding theory has so many coding efficiencycriteria. You wrote about CG (code gain), Quotient of effectiveness and others. Why can't we pick just one?
The answer is 28.
There should not be the only criterion. View an example "Technical Specifications" for the coding system, which can be found at the first main page of our portal (it is in Russian segment of the main page). Pay attention to how many parameters codes and channels involved in such a TS. They should all be selected and matched so that the characteristics of the system were balanced with each other and quite optimal. They should be viewed from different perspectives and taking into account the many features of the designed digital system. The criteria that you specified, also help to designers in the creation TS and its subsequent implementation.
But your question was very timely for us. We have recently formulated another criterion for the efficiency of coding, which was useful to introduce for decoding algorithms operating near the channel capacity. It takes into account the proximity of the working area of the error correction algorithm to the channel capacity C, and the veracity of its decoding. It is most useful to use it just for AWGN channels. Let's call it a quality indicator of the decoding by Zolotarev (QIDZ) for the analyzed algorithms. It is very convenient due to commonly the logarithmic (strongly curved!) scalesfor the energetic parameters of codes are too rough and one-sided. This is particularly noticeable in the areas of high noise channel, i.e., when R≲C.
First, let us formulate this criterion. Let it is given energy of the channel E0 at a certain value of capacity C. Let further for the certain decoding algorithm with error probability of decoding per bit Pb(e) operating energy of the channel is equal to Ew=a*E0 , a>1. For example, if for the obvious condition R<C the algorithm exceeded its energy by 1 dB the level of Shannon's bound, then obviously a=1,26. Further, for a given probability Pb(e) of the evaluated algorithm it will find the required transmission energy without encoding Ex=b*E0 . The value of b may be even greater than 10, if the decoding algorithm really provides a high final veracity. We emphasize that we are now discussing the multiplicity of the energy ratios, not their logarithms, i.e., not decibels. Then the desired QIDZ is defined as
Q=(Ex-E0) / (Ew-E0)=(b-1)/(a-1).
Interpretation of this indicator is very simple: it is the ratio of distance to energy of the Shannon's bound for energy transmission without coding and with coding for the same probability Pb(e) . QIDZ is a very natural criterion for the quality of the decoder: when the decoder's working region tends to the Shannonbound then a -->1, the denominator tends to 0 and Q icreasses to infinity, which is very good, but very difficult and really impossible. And with the increase resulting reliability of decoding b also is growing significantly as QIDZ. Thus, Q in a natural way reflects the quality of the decoding as for the level of working energy, and so reliability.
For quite effective and achievable MTD decoders QIDZ level may exceed Q=40. As for the classic Viterbi algorithm (VA) and a code with K=7 is Q<10. It is also not surprising for such a very short code.
So the number of ways of assessing codes is growing and it is very useful.