Basic Concepts of Encoding Codes and Error Correction
Encoding • Encoding is a transformation procedure operating on the input signal prior to its entry into the communication channel. This procedure adapts the input signal to the communication system and improves its efficiency .
Encoding • In other words, encoding is a procedure for associating words constructed from a finite alphabet of a language (e.g. a natural language) with given words of another language (encoding language) in one-to-one manner. • Decoding is the inverse operation: restoration of words from the initial language.
Codes • Let be the alphabet and its cardinality is . • Any finite sequence of the letters from this alphabet forms a word over it. Let S be a set of all possible words over A. • Some of may be meaningful, some of them may not be meaningful, but anyway we will use only some to encode the information.
Codes • A subset , which is used for representation of the information in the communication system is commonly referred to as the code. • If all words from V have the same length n, then the code V is called the uniform code. • If words from V may have different length, then the code V is called the non-uniform code.
Digital communications • Let us consider the digital communication channel. • Hence . • We will consider the uniform codes of the length n. Thus, the words over Z2 are n -dimensional binary vectors • form a set of “encoding” words.
Distance between binary vectors • The distanceρ(the Hamming distance) between two n -dimensional binary vectors and is the number of components that differ from each other in terms of component-wise comparison. • To find the distance between two binary vectors, it is necessary to add them component-wise by mod 2 and then to count the number of “1s” in the vector-sum.
Distance between binary vectors • For example,
Distance between binary vectors • The distance ρmeets all metric’s axioms: • The Hamming’s norm of a binary vector is the number of “1s” in this vector.
Errors • Replacement of one letter in a word by another one is commonly referred to as the error. • Let the recipient (the receiver) of the information knows the code. • “Detection of the error” means detection of the fact that the error has occurred without the exact detection of where. • “Correction of the error” means the complete restoration of a word, which was originally sent, but then was distorted.
Errors • If the word was transmitted and some bits in X were inverted. As a result, the receiver receives . • If , then the error can not be detected and corrected without analysis of the sense of a whole message. • If , but , then the error can be detected and corrected upon certain conditions.
Maximum likelihood decoding • Let X was transmitted, Y was received and • To correct the error (errors) and to decode the corresponding word we have to find • This method is called the maximum likelihood decoding
Minimum encoding distance • is called a minimum encoding distance of the code V . • This means that • In other words, the minimum encoding distance equals to the minimum distance between the encoding vectors • if the distance between the encoding vector and another vector is less then d then
Minimum encoding distance • For example, let n=3. Then S={000,001,010,011,000,101,110,111}. Let V={000,111}. Then d=3. Indeed, If
Criterion of Error Detection • Theorem. The uniform code V detects at most t errors if, and only if d=t+1. • Proof. Let d=t+1. Let XY and . This means that and t errors can be detected. Let the code detects t errors. Then . Otherwise, if and , this contradicts to the ability to detect t errors.
Example of Error Detection • For example, let n=3. Then S={000,001,010,011,000,101,110,111}. Let V={000,111}. Then d=3 and we can detect (not correct, just detect!!!) 2 errors. • Indeed, if any 1 or 2 of 3 bits in any encoding vector is (are) inverted, we obtain a vector, which does not belong to V. If 3 bits are inverted, we obtain another encoding vector and can not detect the errors.
Criterion of Error Correction • Theorem. The uniform code V can correct at most t errors if, and only if d=2t+1. • Proof. Necessity. The code corrects at most t errors. We have to prove that Suppose that this is not true, which means: Then, if was transmitted, was received and , we are unable to decode, because X and Y are equidistant to Z. This contradicts to the initial condition that the code corrects up to t errors.
Criterion of Error Correction • If was transmitted, was received and ,which means that Y was transmitted. This contradicts to the initial condition that the code corrects up to t errors and therefore it can not be that and this means that and therefore d=2t+1.
Criterion of Error Correction • Proof. Sufficiency. Let the minimum encoding distance is d=2t+1. We have to prove that the code can correct up to t errors. Let was transmitted and was received. Suppose that . On the other hand, according to the metric axioms: If exactly t errors occurred, than X will be decoded. If less then t errors occurred, then, a fortiori, X will be decoded.
Example of Error Correction • For example, let n=3. Then S={000,001,010,011,000,101,110,111}. Let V={000,111}. Then d=3=2*1+1, and we can correct 1 error. • Indeed, if 1 of 3 bits in any encoding vector is inverted, we obtain a vector, which does not belong to V, and we always can determine a unique vector from V, whose distance to the distorted vector is exactly 1.
Example of Error Correction • S={000,001,010,011,000,101,110,111}. V={000,111}. X1=(000) X2=(111) • Let X1=(000) was transmitted, Y=(100) was received. and we definitely decode X1.
Example of Error Correction • S={000,001,010,011,000,101,110,111}. V={000,111}. X1=(000) X2=(111) • If 2 of 3 bits in any encoding vector are inverted, we also obtain a vector, which does not belong to V . We can detect that 2 errors occurred, but we can not correct them, because there will be more than one equidistant vector in V, whose distance to the distorted vector is 2. • Let X1=(000) was transmitted, Y=(101) was received. and there is no way to correct the errors because the decoding procedure can not be ambiguous.
