ECE4253 Digital Communications | |
Department of Electrical and Computer Engineering - University of New Brunswick, Fredericton, NB, Canada | |
The mathematics of error control can be based on either a matrix or a polynomial approach. This page shows how any polynomial G(x) may be used to define an equivalent check matrix and generator matrix. Conversely, it is not always possible to find a polynomial G(x) corresponding to an arbitary generator matrix.
The generator polynomial G(x) can be up to degree p=36, and the input data size is limited to k=36 bits.
G(x) = x5+x4+x2+1
(110101)
This polynomial G(x) is degree 5, giving a 5-bit parity field. (See factors of G(x))
The systematic 15-bit codewords will have 10 data bits and 5 parity bits.
When codewords are cyclic the circular shift of a valid codeword produces another valid codeword.
For example, rotating the 15-bit codeword (01) left by one bit gives the codeword (02): (02) = 000000001101010 |
DATA = 00 : | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
DATA = 01 : | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 |
DATA = 02 : | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
DATA = 03 : | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 |
DATA = 04 : | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
DATA = 05 : | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 |
DATA = 06 : | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
DATA = 07 : | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |
DATA = 08 : | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
DATA = 09 : | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
DATA = 10 : | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |
DATA = 11 : | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
DATA = 12 : | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 |
DATA = 13 : | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
DATA = 14 : | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
DATA = 15 : | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 |
When codewords are linear, any linear combination of codewords is another codeword. For example, the 15-bit codeword (01) is the sum (02)+(03) (03) = 000000001101010 (01) = 000000000110101 |
This sample subset of 15-bit codewords has a minimum distance D=4, correcting up to t=1 error.
00 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | |
00 | -- | 04 | 06 | 04 | 04 | 06 | 04 | 04 | 04 | 04 | 04 | 06 | 06 | 04 | 04 | 08 |
01 | 04 | -- | 04 | 06 | 06 | 04 | 04 | 04 | 04 | 04 | 06 | 04 | 04 | 06 | 08 | 04 |
02 | 06 | 04 | -- | 04 | 04 | 04 | 04 | 06 | 04 | 06 | 04 | 04 | 04 | 08 | 06 | 04 |
03 | 04 | 06 | 04 | -- | 04 | 04 | 06 | 04 | 06 | 04 | 04 | 04 | 08 | 04 | 04 | 06 |
04 | 04 | 06 | 04 | 04 | -- | 04 | 06 | 04 | 06 | 04 | 04 | 08 | 04 | 04 | 04 | 06 |
05 | 06 | 04 | 04 | 04 | 04 | -- | 04 | 06 | 04 | 06 | 08 | 04 | 04 | 04 | 06 | 04 |
06 | 04 | 04 | 04 | 06 | 06 | 04 | -- | 04 | 04 | 08 | 06 | 04 | 04 | 06 | 04 | 04 |
07 | 04 | 04 | 06 | 04 | 04 | 06 | 04 | -- | 08 | 04 | 04 | 06 | 06 | 04 | 04 | 04 |
08 | 04 | 04 | 04 | 06 | 06 | 04 | 04 | 08 | -- | 04 | 06 | 04 | 04 | 06 | 04 | 04 |
09 | 04 | 04 | 06 | 04 | 04 | 06 | 08 | 04 | 04 | -- | 04 | 06 | 06 | 04 | 04 | 04 |
10 | 04 | 06 | 04 | 04 | 04 | 08 | 06 | 04 | 06 | 04 | -- | 04 | 04 | 04 | 04 | 06 |
11 | 06 | 04 | 04 | 04 | 08 | 04 | 04 | 06 | 04 | 06 | 04 | -- | 04 | 04 | 06 | 04 |
12 | 06 | 04 | 04 | 08 | 04 | 04 | 04 | 06 | 04 | 06 | 04 | 04 | -- | 04 | 06 | 04 |
13 | 04 | 06 | 08 | 04 | 04 | 04 | 06 | 04 | 06 | 04 | 04 | 04 | 04 | -- | 04 | 06 |
14 | 04 | 08 | 06 | 04 | 04 | 06 | 04 | 04 | 04 | 04 | 04 | 06 | 06 | 04 | -- | 04 |
15 | 08 | 04 | 04 | 06 | 06 | 04 | 04 | 04 | 04 | 04 | 06 | 04 | 04 | 06 | 04 | -- |
To determine the minimum distance between any two codewords in a linear block code, it is sufficient to check every codeword once against the all-zero codeword. In other words, the Hamming distance of a code may be determined from the distances in row 00 only. Moreover, the distance from a given codeword to zero is found by the sum of the 1's in the codeword. The remaining codewords could be easily checked in this way. |
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(8,7) Simple Parity Bit (D=2) no error correction
(7,4) Hamming Code (D=3) single bit error correction
(15,11) Hamming Code (D=3) single bit error correction
(15,10) Extended Hamming Code (D=4) single bit error correction
(31,21) BCH Code (D=5) double bit error correction (notes)
(15,5) BCH Code (D=7) triple bit error correction (notes)
(23,12,7) Binary Golay Code (D=7) triple bit error correction
(35,27) Fire Code specialized 3-bit burst error correction
16-bit CRC (CCITT) commonly used for error detection (notes)
2024-04-29 15:55:12 ADT
Last Updated: 2015-02-06 |
Richard Tervo [ tervo@unb.ca ] | Back to the course homepage... |