Reed-Solomon | Reed Solomon BCH encoder and decoder | Messaging library

miscorrection for error-only decoding comes from computing the wrong error locations

Yes. For example, consider an RS code with 6 parities, which can correct 3 errors. Assume that 4 errors have occurred, and that a 3 error correction attempt created an additional 3 errors, for a total of 7 errors. It will produce a valid codeword, but the wrong codeword.

There are situations where the probability of miscorrection can be lowered. If the message is a shortened message, say 64 bytes of data and 6 parities for a total of 70 bytes, then if a 3 error case produces an invalid location, miscorrection can be avoided. In this case, the odds of 3 random locations being valid is (70/255)^3 ~= .02 (2%).

Another way to avoid miscorrection is to not use all of the parities for correction. With 6 parities, the correction could be limited to 2 errors, leaving 2 parities for detection purposes. Or use 7 parities, for 3 error correction, with 1 parity used for detection.

Follow up based on comments:

First note that there are 3 decoders that can be used for BCH view Reed Solomon: PGZ (Peterson Gorenstein Zierler) matrix decoder, BKM (Berlekamp Massey) discrepancy decoder , and Sugiyama's extended Euclid decoder. PGZ has greater time complexity O((n-k)^3) than BKM or Euclid, so most implementations use BKM or Euclid. You can read a bit more about these decoders here:

https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction

Getting back to 6 parities, 4 errors. All valid RS(n+6, n) codewords differ from each other by at least 7 elements. If 4 of the elements of a message are in error, there may be a valid codeword that differs by only 3 more elements from that message with 4 error elements, and in this case, all 3 decoders will find that the message differs from a valid codeword by 3 elements, and "correct" those 3 elements to produce a valid codeword, but in this case the wrong valid codeword, which will differ by 7 elements from the original codeword. With 5 elements in error, a 2 or 3 error miscorrection could occur, and with 6 or more elements in error, a 1 or 2 or 3 error miscorrection could occur.

Invalid location - consider a RS code based on GF(2^8), which allows for a message size up to 255 bytes. Valid locations for a 255 byte message are 0 to 254. If the message size is less than 255 bytes, for example 64 data + 6 parity = 70 bytes, then locations 0 to 69 are valid, while locations 70 to 254 are invalid. In what would otherwise be a case of miscorrection, if a calculated location is out of range, then the decoder has detected an uncorrectable message, rather than miscorrect it. Assume a garbled message and that the decoder generates 3 random locations in the range 0 to 254, the probability of all 3 being in the range 0 to 69 is (70/255)^3.

Another case where miscorrection is avoided is when the number of distinct roots of the error locator polynomial does not match the degree of the polynomial. Consider a 3 error case with generated error locator polynomial x^3 + a x^2 + b x + c. If there are more than 3 errors in the message, then the generated polynomial may have less than 3 distinct roots, such as a double root, or zero roots, or ... , in which case miscorrection is avoided and the message is detected as being uncorrectable.

The encoding matrix is a 6 x 4 Vandermonde matrix using the evaluation points {0 1 2 3 4 5} modified so that the upper 4 x 4 portion of the matrix is the identity matrix. To create the matrix, a 6 x 4 Vandermonde matrix is generated (where matrix[r][c] = pow(r,c) ), then multiplied with the inverse of the upper 4 x 4 portion of the matrix to produce the encoding matrix. This is the equivalent of "systematic encoding" with Reed Solomon's "original view" as mentioned in the Wikipedia article you linked to above, which is different than Reed Solomon's "BCH view", which links 1. and 2. refer to. The Wikipedia's example systematic encoding matrix is a transposed version of the encoding matrix used in the question.

https://en.wikipedia.org/wiki/Vandermonde_matrix

https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction#Systematic_encoding_procedure:_The_message_as_an_initial_sequence_of_values

The code to generate the encoding matrix is near the bottom of this github source file:

https://github.com/Backblaze/JavaReedSolomon/blob/master/src/main/java/com/backblaze/erasure/ReedSolomon.java

An important aspect is that Reed-Solomon (RS) codes are cyclic: the set of codewords is stable by cyclic shift.

A consequence is that no particular part of a code word is more protected or less protected.

A RS code has a error correction capability equal to t = (n-k)/2, where n is the code length (generally expressed in bytes) and k is the information part length.

If the total number of errors (in both parts) is less than t, the RS decoder will be able to correct the errors (more precisely, the t erroneous bytes in the general case). If it is higher, the errors cannot be corrected (but could be detected, another story).

The emplacement of the errors, either in the information part or the added part, has no influence on the error correction capability.

EDIT: the rule t = (n-k)/2 that I mentioned is valid for Reed-Solomon codes. This rule is not generally correct for BCH codes: t <= (n-k)/2. However, with respect to your question, this does not change the answer: these families of code have a given capacity correction, corresponding to the minimum distance between codewords, the decoders can then correct t errors, whatever the position of the errors in the codeword

You could use a shortened code word such as (5, 19, 6, 7) 31.5% correction ratio, 95 bits, 48 bars. One advantage of a shortened code word is reduced chance of mis-correction if it is allowed to correct the maximum of 6 errors. If any of the 6 error locations is outside of the range of valid locations, that is an indication of that there are more than 6 errors. The probability of mis-correction is about (19/31)^6 = 5.3%.

Posting the answer :

First, I made a mistake in the modulo-2 algebra used to expand the polynomial. Non-modulo expanded form is :