Run length limited or RLL coding is a line coding technique that is used to send arbitrary data over a communications channel with bandwidth limits. This is used in both telecommunication and storage systems which move a medium past a fixed head. Specifically, RLL bounds the length of stretches (runs) of repeated bits during which the signal does not change. If the runs are too long clock recovery is difficult, while if they are too short, the high frequencies might be attenuated by the communications channel. By modulating the data, RLL reduces the timing uncertainty in decoding the stored data which would lead to the possible errant insertion or removal of bits when reading the data back. Run length limited codes were widely used in hard disk drives until the mid-1980s and are used in digital optical discs such as CD, DVD, MD, Hi-MD and Blu-ray as the mechanism to ensure that the boundaries between bits can always be accurately found (preventing bit slip) while efficiently using the media to reliably store the maximum amount of data in a given space. Early disk drives used very simple encoding schemes, such as RLL (0,1) FM code, but higher density RLL (2,7) and RLL (1,7) codes became the de facto industry standard for hard disks by the early 1990s.
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On a hard disk, information is represented by changes in the direction of the magnetic field on the disk. In a computer, information is represented by the voltage on a wire. No voltage on the wire would be a binary zero, and a positive voltage on the wire represents a binary one. Magnetic media, on the other hand, always carries a magnetic flux - either a "north" pole or a "south" pole. In order to convert the magnetic fields to binary data, some encoding method must be used to translate between the two.
One of the simplest practical codes, Modified Non-Return-to-Zero-Inverted (NRZI), simply encodes a 1 as a magnetic polarity transition, also known as a "flux reversal", and a zero as no transition. With the disk spinning at a constant rate, each bit is given an equal time period, a "data window," for the magnetic signal that represents that bit, and the flux reversal, if any, occurs at the start of this window. (Note: older hard disks used the same data window over the whole disk, but modern disks are more complicated; for more on this, see zoned bit recording.)
While this method is simple, it is prone to errors for long runs of zeros.
In a simple example, consider the binary pattern 101 with a data window of 1 ns (one nanosecond, or one billionth of a second). This will be stored on the disk as a change, followed by no change, and then another change. If the preceding magnetic polarity was already positive, the resulting pattern might look like this: −−+. A value of 255, or all binary ones, would be written as −−+−+−+−+ or +−+−+−+−. A zero byte would be written as ++++++++ or −−−−−−−−. A 512 byte sector of zeros would be written as 4,096 sequential bits with the same polarity.
Since a disk drive is a physical piece of hardware, the rotational speed of the drive can change slightly, due to a change in the motor speed or thermal expansion of the disk platter. The physical media on a floppy disk can also become deformed, causing larger timing errors, and the timing circuit on the controller itself may have small variations in speed. The problem is that, with a long string of zeros, there's no way for the disk drive's controller to know the exact position of the read head, and so a speed variation of even .01% could result in 4 bits being added to or removed from the data stream. Without some form of synchronization and error correction, the data would become completely unusable.
The other problem is due to the limits of magnetic media itself: it is only possible to write so many polarity changes in a certain amount of space, so there's an upper limit to how many 1's can also be written sequentially.
To prevent this problem, data is coded in such a way that long repetitions of a single binary value do not occur. By limiting the number of zeros written consecutively, this makes it possible for the drive controller to stay in sync. By limiting the number of 1's written in a row, the overall frequency of polarity changes is reduced, allowing the drive to store more data in the same amount of space, resulting in either a smaller package for the same amount of data or more storage in the same size package.
All codes used to record on magnetic disks have limited the run of zero transitions and can therefore be characterized as RLL codes. In the early years they were given specific names, such as Modified Frequency Modulation (MFM).
The RLL terminology in hard disk drive art, specifically RLL (2,7), was originally developed by IBM engineers, and was first used commercially in 1979 on the IBM 3370 DASD,[1][2][3] for use with the 4300 series mainframe. During the late 1980s, PC hard disks began using RLL. RLL codes have found almost universal application in optical disc recording practice since 1980. In consumer electronics, RLLs like the EFM code with (Eight-to-Fourteen:rate = 8/14, d=2, k=10) are employed in the Compact Disc (CD) and MiniDisc (MD), and the EFMPlus code (rate = 8/16, d=2, k=10) used in the DVD. Parameters d, k are the minimum and maximum allowed run-lengths. For more coverage on the storage technologies, the references cited in this article are useful.
Generally run-length is the number of bits for which signal remains unchanged. A run-length of 3 for bit 1, represents a sequence of '111'. For instance, the pattern of magnetic polarizations on the disk might be '+−−−−++−−−++++++', with runs of length 1, 4, 2, 3, and 6. However, run length limited coding terminology assumes NRZI encoding, so 1 bits indicate changes and 0 bits indicate the absence of change, the above sequence would be expressed as '11000101001000001', and only runs of zero bits are counted.
Somewhat confusingly, the run length is the number of zeros (0, 3, 1, 2 and 5 in the preceding) between adjacent ones, which is one less than the number of bit times the signal actually remains unchanged. Run length limited sequences are characterized by two parameters, (d) and (k), which stipulate the minimum and maximum zero-bit run length that can occur in the sequence. So RLL's are generally specified as (d,k) RLL. e.g.: (1,3) MFM RLL.
Suppose a magnetic tape can support up to 3,200 flux reversals per inch. A Modified Frequency Modulation or (1,3) RLL encoding stores each data bit as two bits on tape, but since there is guaranteed to be one 0 (non flux reversal) bit between any 1 (flux reversal) bits then it is possible to store 6,400 encoded bits per inch on the tape, or 3,200 data bits per inch. A (1,7) RLL encoding can also store 6,400 encoded bits per inch on the tape, but since it only takes 3 encoded bits to store 2 data bits this is 4,267 data bits per inch. A (2,7) RLL encoding takes 2 encoded bits to store each data bit, but since there is guaranteed to be two 0 bits between any 1 bits then it is possible to store 9,600 encoded bits per inch on the tape, or 4,800 data bits per inch.
The flux reversal densities on hard drives are significantly greater, but the same improvements in storage density are seen by using different encoding systems.
In the encoded format a "1" bit indicates a flux transition, while a "0" indicates that the magnetic field on the disk does not change for that time interval.
Generally, the term "RLL code" is used to refer to more elaborate encodings, but the original frequency modulation code, also called differential Manchester, can be seen as a simple rate-1/2 RLL code. The added 1 bits are referred to as clock bits.
| Data | Encoded |
|---|---|
| 0 | 10 |
| 1 | 11 |
Example:
Data: 0 0 1 0 1 1 0 1 0 0 0 1 1 0 Encoded: 1010111011111011101010111110 Clock: 1 1 1 1 1 1 1 1 1 1 1 1 1 1
By extending the maximum run length to 2 adjacent 0 bits, the data rate can be improved to 4/5. This is the original IBM group code recording variant.
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Example:
Data: 0010 1101 0001 1000 Encoded: 10010011011101111010
Modified frequency modulation begins to get interesting, because it imposes a constraint of d=1 (a minimum of one 0 between each two 1's), thereby allowing a bit recording rate equal to twice the flux reversal limit. Although its coding rate is only 1/2, the doubled recording rate limit makes it equivalent to a rate-1 code.
| Data | Encoded |
|---|---|
| 0 | x0 |
| 1 | 01 |
Where "x" is the complement of the previous data bit (which is also the previous encoded bit). Except for that x bit, this is the same as the FM table, which is how this code gets its name. The inserted clock bits are 0 except between two 0 data bits.
Example:
Data: 0 0 1 0 1 1 0 1 0 0 0 1 1 0 Encoded: x010010001010001001010010100 Clock: x 1 0 0 0 0 0 0 0 1 1 0 0 0
(1,7) RLL maps 2 bits of data onto three bits on the disk, and the encoding is done in two or four bit groups. The encoding rules are: (x, y) becomes (NOT x, x AND y, NOT y), except (x, 0, 0, y) becomes (NOT x, x AND y, NOT y, 0, 0, 0) [4]
| Data | Encoded |
|---|---|
| 00 00 | 101 000 |
| 00 01 | 100 000 |
| 10 00 | 001 000 |
| 10 01 | 010 000 |
| 00 | 101 |
| 01 | 100 |
| 10 | 001 |
| 11 | 010 |
Example:
Data: 0 0 1 0 1 1 0 1 0 0 0 1 1 0 Encoded: 101 001 010 100 100 000 001
(2,7) RLL maps n bits of data onto 2n bits on the disk, but the encoding can be done in two, three or four bit groups.
| Data | (2,7) RLL Encoded |
|---|---|
| 11 | 1000 |
| 10 | 0100 |
| 000 | 000100 |
| 010 | 100100 |
| 011 | 001000 |
| 0011 | 00001000 |
| 0010 | 00100100 |
Example:
Data: 1 1 0 1 1 0 0 1 1 Encoded: 1000 001000 00001000
There is also an alternate (1,7) RLL encoding that is sometimes used to avoid a DC bias (which helps when sending a signal over a long distance or with some types of recording media).
| Data | Encoded |
|---|---|
| 00 | x01 |
| 01 | 010 |
| 10 | x00 |
| 11 00 | 010 001 |
| 11 01 | x00 000 |
| 11 10 | x00 001 |
| 11 11 | 010 000 |
Where "x" is the complement of the previous encoded bit (i.e. 1 if the previous bit was 0, and 0 if the previous bit was 1).
Example:
Data: 0 1 0 0 1 1 0 1 0 1 Encoded: 010 101 000 000 010
This article was originally based on material from the Free On-line Dictionary of Computing, which is licensed under the GFDL.