GB2412453A

GB2412453A - 6/3 Hamming Error Correction

Info

Publication number: GB2412453A
Application number: GB0407002A
Authority: GB
Inventors: David Hostettler Wain
Original assignee: Individual
Current assignee: Individual
Priority date: 2004-03-27
Filing date: 2004-03-27
Publication date: 2005-09-28
Also published as: GB0407002D0

Abstract

A 6/3 hamming error correction technique has a hamming conversion table which converts all eight possible values of a three bit wibble into a corresponding six bit reference wobble. Each wobble reference value is at least three bits different from the other seven. A dehamming conversion table may be generated from the hamming conversion table whereby single bit errors (values which differ from the reference values by one bit) may be corrected back to the reference value and the single bit error corrected. Normally such a process would be automated by computer, converting a stream or packet of hammed data back into raw form. A periodic or packet checksum should be added to the raw data so that multiple bit errors may be detected.

Description

6/3 HAMMING ERROR CORRECTION This invention relates to a digital error

correction technique known as 6/3 hamming.

Hamming is a well known technique of protecting digital data from errors. The simplest and best known is 8/4 hamming. 8/4 hamming converts four bit binary nibbles into eight bit bytes, doubling the amount of data. However, the hammed data is error tolerant.

The 8/4 hamming process normally uses a hamming conversion table for the sixteen possible four bit nibbles containing corresponding eight bit byte reference values. Each byte reference value differs from the other fifteen by at least four bits.

During hamming, each four bit nibble of an input stream or packet of digital data is converted to its eight bit reference byte. The output stream or packet (of hammed data) is hence twice the size of the input.

Errors may be corrected because if a bit in a byte becomes errored, the closest value as defined by the number of bit flips is used. Thus single bit errors (within an eight bit hammed byte) are detected and correctable. One bit error per eight hammed bits is correctable (A 12%).

Double bit errors are detected but the errored byte is equi-distant from at least two byte reference values. Hence, double bit errors (per byte) are detected but cannot be corrected.

Dehamming usually uses a 8/4 dehamming conversion table for the 256 possible byte values, of which only 16 are the valid reference entries. There are 128 single bit error entries (correctable) and 112 multiple bit error entries (not correctable).

8/4 hamming can only corre* single bit errors. However, in order to do this, each reference value need only be three bits different from the others, rather than four.

Multiple bit errors should be detected (but not corrected) by a checksum anyway. - 2

According to the present invention there is provided a new hamming technique that detects and corrects one bit error in every six hammed bits, with optional detection of double or higher bit errors by a checksum.

A s,oecific embodiment of the invention wilt now be described by way of example with reference to the accompanying figures: Figure 1 shows two 6t3 hamming binary tables.

Figure 2 shows a 6/3 dehamming hexadecimal array for table 2.

this new hamming technique is known as 6/3 hamming. Three bit wibbles are converted into six bit wobbles using a conversion table. Each wobble reference value is at least THREE BITS DIFFERENT from the other seven reference values.

If a bit error occurs within a wobble, it may be recovered to its nearest reference value, as defined by the number of bit flips. It may then be converted back to its prehammed wibble. Thus a single bit error (SBE) may be tolerated in every six bits (A 17%) of hammed data However, it does not detect many double bit errors, but since these are unrecoverable anyway, they are best detected using a checksum. The amount of data is still doubled but the error resilience is better than 8/4 hamming.

There are many 6/3 hamming conversion tables which may be found by a trivial process of finding eight six bit values that are different from each other by three or more bits. Of the two tables shown in figure 1, table 2 is favoured, because it has a better distribution of Os and Is. In some applications this could be significant.

Dehamming usually uses a 6/3 dehamming conversion table for the 64 possible wobble values, of which only 8 are valid reference entries. There are 48 single bit error entries (correctable and 8 multiple bit error entries (not correctable). The 6/3 dehamming conversion table may be generated by treating each of the 64 possible 1) an exact match if it matches a reference value; or 2) a single bit error if it is one bit different from a reference value; or 3) a multiple bit error.

In applications where it is expected that bit errors occur together packets should be interleaved after hamming. By interleaving bits, systematic errors are avoided. For example, each sixth bit should be put into its corresponding sixth of a packet (of data).

Hence, Wobble 00000011 11112222 22333333 44444455 5555 Bit 54321054 32105432 10543210 54321054 3210 Becomes Wobble 01234501 23450123 45012345 01234501 2345 Bit 55555544 44443333 33222222 11111100 0000

APPENDIX A

ERROR CORRECTION PROBABILITIES

Assuming no systematic effects, in order to determine the probabilities of packets of data being correctable the following derivations are provided. Let,

Pe be the probability of a single bit in error Np be the number of bits in a packet n be the number of bit errors in a packet The probability of one combination of exactly n bit errors occurring is: n Np - n Po = Pe. (1Pe) If we have a combination of n error bits, the probability that the (m+1)th bit is in a previously errored wobble is the same as that of the mth bit, with an extra term: Pw(m+1) = Pw(m) + 5 (1 - Pw(m) ) ; with Pw(1) - 0 (Np- m) Hence probability of this combination being non- correctable (at least one wobble containing two or more bit errors) is: n- 1 \ Pc = / Pw(m+1); the sum of Pw(m+1) from m=0 to m=n-1 / m=0 The number of combinations of n bit errors in a packet of Np bits is: Nc = Np! n! (Np n)! - 4 Thus the probability of a packet occurring with exactly n bit errors that is non- correctable (two or more of the bits in at least one wobble) is: n-1 n Np - n \ Pn = Po. Nc. Pc = Pe, (1- Pe) . Np! ./ Pw(m+1) n! (Np- n)! / m=0 For completeness, the probability of a packet with n bit errors occurring is: n Np- n Pr = Po. Nc = Pe. (1- Pe) . NO! n! (Np- n)!

Claims

CI-AIMS

1. A digital error correction technique comprising a hamming table of conversion values, converting three bit wibble values into six bit wobble reference values.

2. A digital error correction technique as claimed in Claim 1 wherein each six bit wobble reference value is at least 3 bits different from the seven others.

3. A digital error correction technique as claimed in Claims and 2 wherein interleaving the bits may be used to correct any systematic errors.

4. A digital error correction technique as claimed in Claims 1, 2 and 3 wherein a corresponding dehamming table of conversion values converting six bit wobbles back into three bit wibbles may be used to correct a single bit error per six bit

5. A digital error correction technique as claimed in Claims 1, 2, 3 and 4 wherein two or more bit errors per six bit wobble may be detected by a checksum but not corrected.