A certain telemetry message consists of 960 binary digits, 40% of them being zeros, randomly distributed. Transmitted thru not-perfectly-reliable media, on the average 35% of "ones" become zeroes and 20% of zeroes become "ones".

1. What is the probability of randomly chosen bit to arrive undistorted?

A self-correcting algorithm is applied adding 4 additional bits to each 3-bit "character"(see Hamming on the web). The new block is transmitted through the same faulty media, decoded to get the message only and the same question is asked:

2.What is the probability of randomly chosen bit to arrive undistorted?

Rem: If solved by simulation, define the achieved accuracy. Your ideas about improving the "reliability figure" will be welcome.

Based on a reading of Wikipedia:

Due to the limited redundancy that Hamming codes add to the data, they can only detect and correct errors when the error rate is low. The current example looks inappropriate for Hamming, since the probability that no data in a given 7- bit block is inaccurate is 0.71^7, or less than 10%. Another potential issue is that Hamming assumes a binary symmetric channel where errors are equally likely for 1 and 0, which is not the case here.

Further, Hamming (7.4) only allows the correction of 1 bit of data per block.  If error correction is performed on a two-bit error the result will be garbled. Of the 240 blocks of data, around 10% will be error-free anyway, and another 12% will have just one error.
This means that in 78% of cases the data will be inaccurate after correction, i.e. 52 blocks will be received correctly, while 188 blocks will be garbled by error correction.

In addition there is the relatively low probability that errors in the data and check bits will compound to give an apparently accurate block. I have not computed this.

I assume this is what the title, IMPROVED?, is getting at.

Posted by broll on 2018-04-28 00:28:39