GB2413652A

GB2413652A - 7/4 Hamming error correction

Info

Publication number: GB2413652A
Application number: GB0409430A
Authority: GB
Inventors: David Hostettler Wain
Original assignee: Individual
Current assignee: Individual
Priority date: 2004-04-28
Filing date: 2004-04-28
Publication date: 2005-11-02
Also published as: GB0409430D0

Abstract

A 7/4 hamming error correction technique has a hamming conversion table which converts all sixteen possible values of a four bit nibble into a corresponding seven bit reference bobble. Each bobble reference value is at least three bits different from the other fifteen. 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

24 1 3652 - 1 7/4 HAMMING ERROR CORRECTION This invention relates to a

digital error correction technique known as 7/4 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 data3 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 ( 12%).

Double bit errors are detected but the errored byte is equi-diseant 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 (uncorrectable).

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.

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

A specific embodiment of the invention will now be described by way of example with reference to the accompanying figures: Figure 1 shows 7/4 hamming table 1.

Figure 2 shows 7/4 hamming table 2.

Figure 3 shows 7/4 dehamming table t.

Figure 4 shows 7/4 dehamming table 2.

This new hamming technique is known as 7/4 hamming. Four bit nibbles are converted into seven bit bobbles using a conversion table. Each bobble reference value is at least THREE BITS DIFFERENT from the other fifteen reference values.

If a bit error occurs within a bobble, 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 nibble. Thus a single bit error (SBE) may be tolerated in every seven bits (A 14%) of hammed data.

However, it does not detect double bit errors, but since these are unrecoverable anyway, they are best detected using a checksum. The error resilience is better than 8/4 hamming and the amount of data is only increased by 75%.

Also, even compared with 6/3 hamming (see my earlier patent) although the error resilience appears worse, the amount of data is not doubled. The amount of hammed data is 12.5% less than 8/4 or 6/3 hamming, reducing packet size which reduces the chance of error.

There are many 7/4 hamming conversion tables which may be found by a trivial process of finding sixteen seven bit values that are different from each other by three or more bits. Of the two hamming tables shown, 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 7/4 dehamming conversion table for the 128 possible bobble values, of which only 16 are valid reference entries. There are 112 single bit error entries, all of which are classed as correctable. The 7/4 dehamming conversion table may be generated by treating each of the 128 possible bobble values as: 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.

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 seventh bit should be put into its corresponding seventh of a packet (of data).

Hence, Bobble 00000001 11111122 22222333 33334444 44455555 55666666 67777777 Bit 65432106 54321065 43210654 32106543 21065432 10654321 06543210 Becomes Bobble 01234560 1234560123456012 34560123 45601234 56012345 60123456 Bit 77777776 66666655 55555444 44443333 33322222 22111111 10000000

APPENDIX A

ERROR CORRECTION PROBABILITIES

Assuming no systematic effects, in order to determine the probabilities of packets of 7/4 hammed 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. (1- Pe) If we have a combination of n error bits, the probability that the (m+1)th bit is in a previously errored bobble is the same as that of the mth bit, with an extra term: Pw(m+1) = Pw(m) + 6 (1 - Pw(m) ) ; with Pw(1) = 0 (Npm) Hence probability of this combination being non-correctable (at least one bobble containing two or more bit errors) is: n \ Pc = / Pw(m+l); the sum of Pw(m+1) from m=0 to m=n-1 l m=0 The number of combinations of n bit errors in a packet of Np bits is: Nc = Np! n! (Np- n)! Thus the probability of a packet occurring with exactly n bit errors that is noncorrectable (two or more of the bits in at least one bobble) is: n-1 n Npn \ Pn = Po. Nc. Pc = Pe. (1- Pe) . Np! . / Pw(m+1) n! (Np- n)! / m=0 - 4 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

CLAMS

1. A digital error correction technique comprising a hamming table of conversion values, converting four bit nibble values into seven bit bobble reference values.

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

3 A digital error correction technique as claimed in Claims 1 and 2 wherein interleaving the hammed bits may be used to corre* 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 seven bit bobbles back into four bit nibbles may be used to correct a single bit error per seven bit bobble.

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