Coding theory deals with the properties of codes, and thus with their fitness for a specific application.
There are two classes of codes.
- Source Coding [Entropy Encoding]
- Channel Coding [Error Control Encoding]
The first is Source Encoding which attempts to compress the data from a source in order to transmit it more efficiently. We see this in practice every day on the Internet where the common "Zip" data compression is used to reduce the network load and make files smaller. The second is Channel Encoding. This technique adds extra data bits, commonly called parity bits, to make the tranmission of data more robust to disturbances present on the transmission channel. There are many application that the ordinary user is not aware of that utilize channel encoding. A typical music CD has powerful BCH block codes to correct for scratches and dust. In this application the transmission channel is the CD itself. Cell phones also use powerful coding technique to correct for the fading and noise of high frequency radio transmission. Data modems, telephone transmission and of course NASA all employ powerful channel coding to get the bits through.
Source Encoding
The aim of source encoding is to take the source data and make it smaller. FAX transmission which has been around for many years uses a simple run length code. The principle is to recognize that most documents are white space with brief interruptions for the black typing. So FAX compresses a document by adding a repeat count to the next transition. It may tell the receiver that 100 of the next pixels are white. Another common encoding technique is string compression. This is used for data files. The encoder has a dictionary of strings. It matches the incoming text to the strings in the dictionary and when found, it will send a single number to the receiver which is the index to the string. All know implementations of string compression are adaptive in nature and allow the encoder to create new strings and transmit them to the decoder so the two dictionaries remain the same.
Channel Encoding
The aim of channel encoding theory is to find codes which transmit quickly, contain many valid code words and can correct or at least detect many errors. These aims are mutually exclusive however, so different codes are optimal for different applications. The needed properties of this code mainly depend on the probability of errors happening during transmission. In a typical CD, the impairment is mainly dust or scratches. Thus codes are used in an interleaved manner. The data is spread out over the disk. Although not a very good code, a simple repeat code can serve as an understandable example. Suppose we take a block of data bits (representing sound) and send it three times. At the receiver we will examine the three repetitions bit by bit and take a majority vote. The twist on this is that we don't merely send the bits in order. We interleave them. The block of data bits is first divided into 4 smaller blocks. Then we cycle through the block and send one bit from the first, then the second, etc. This is done three times to spread the data out over the surface of the disk. In the context of the simple repeat code, this may not appear effective. However, there are more powerful codes known which are very effective at correcting the "burst" error of a scratch or a dust spot when this interleaving technique is used.
Other codes are more appropriate for different applications. Deep space communications are limited by the thermal noise of the receiver which is more of a continuous nature than a bursty nature. Likewise, narrowband modems are limited by the noise present in the telephone network and is also modeled better as a continuous disturbance. Cell phones are troubled by rapid fading. The high frequencies used can cause rapid fading of the signal even if the receiver is moved a few inches. Again there are a class of channel codes that are designed to combat fading.
The term algebraic coding theory denotes the sub-field of coding theory where the properties of codes are expressed in algebraic terms and then further researched.
Algebraic Coding theory, is basically divided into two major types of codes
- Linear Block Codes
- Convolutional Codes
It analyzes the following three properties of a code -- mainly:
- code word length
- total number of valid code words
- the minimum Hamming distance between two valid code words
Linear Block Codes
Linear Block codes, have the property of linearity, i.e the sum of any two codewords is also a code word; and they are applied to the source bits in blocks; hence the name Linear Block Codes.
Any linear block code, is represented at where
- n , is the length of the codeword, in symbols,
- k , is the number of source symbols that will be used for encoding at once,
- , is the minimum hamming distance for the code
There are many types within linear block codes, like
- Cyclic Codes [ Hamming Code, is a subset of cyclic codes]
- Repetition Codes
- Parity Codes
- Reed Solomon Codes
- BCH
- Reed Muller codes
Convolution Codes
Here the idea is to make every codeword symbol, be the weighted sum of the various, input message symbols. This is like Convolution used in LTI systems to find the output of a system, when you know the input and impulse response.
So we generally find the output of the system [convolutional encoder] , which is the convolution of the input bit, against the states of the convolution encoder, registers.
This type of coding, has advantage of using a state-machine model, and can correct more errors.
The viterbi algorithm is used to decode convolutional codes.
Applications of Coding Theory
Another concern of coding theory is designing codes that help synchronization. A code may be designed so that a phase shift can be easily detected and corrected and that multiple signals can be sent on the same channel. There is an interesting class of coded we see every day on our cell phones. These are the Code Division Multiple Access (CDMA) codes. The details are beyond the scope of this discussion but briefly, each phone is assigned a codeword from a special class (algebraic field). When transmitting, the code word is used to scramble the bits representing the voice message. At the receiver, a descrambling process is done to decipher the message. The properties of this class of code words allow many users (with different codes) to use the same radio channel at the same time. The receiver, using the descrambling, will only "hear" other callers as low level "noise".
Another popular class of codes are the Automatic Repeat reQuest (ARQ) codes. In this general class, the transmitter adds the parity check bits to a longer message. The receiver checks the parity bits against the message and if there is not a match, it will ask the transmitter to retransmit the message. Almost all wide area networks WAN and protocols except for the very simple ones use ARQ retransmission. Common protocols include SDLC (IBM), TCP (Internet), X,25 (International) and many others. There is an extensive field of research on this topic because of the problem of matching a rejected packet against a new packet. Is it a new one or is it a retransmission? Typically numbering schemes has been used although in some networks, the packet may have another identifier or it may be left to higher layers to request retransmission. TCP/IP is a good example of a protocol that supports both techniques. In a connected scenario, TCP/IP leaves the retransmission to the network thus it uses the ARQ coding. In a connectionless network, ARQ is not used. Instead, it is up to the application to examine the packet and request retransmission as needed. This may go as high up as requiring the user to hit the "refresh" button on a browser. But, even this is still in the class of ARQ research; the user just has to become involved.