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In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code. They are error-correcting codes that have algebraic properties that are convenient for efficient error detection and correction.

If 00010111 is a valid codeword, applying a right circular shift gives the string 10001011. If the code is cyclic, then 10001011 is again a valid codeword. In general, applying a right circular shift moves the least significant bit (LSB) to the leftmost position, so that it becomes the most significant bit (MSB); the other positions are shifted by 1 to the right.

Contents

DefinitionEdit

Let   be a linear code over a finite field (also called Galois field)   of block length n.   is called a cyclic code if, for every codeword c=(c1,...,cn) from C, the word (cn,c1,...,cn-1) in   obtained by a cyclic right shift of components is again a codeword. Because one cyclic right shift is equal to n − 1 cyclic left shifts, a cyclic code may also be defined via cyclic left shifts. Therefore the linear code   is cyclic precisely when it is invariant under all cyclic shifts.

Cyclic Codes have some additional structural constraint on the codes. They are based on Galois fields and because of their structural properties they are very useful for error controls. Their structure is strongly related to Galois fields because of which the encoding and decoding algorithms for cyclic codes are computationally efficient.

Algebraic structureEdit

Cyclic codes can be linked to ideals in certain rings. Let   be a polynomial ring over the finite field  . Identify the elements of the cyclic code C with polynomials in R such that   maps to the polynomial  : thus multiplication by x corresponds to a cyclic shift. Then C is an ideal in R, and hence principal, since R is a principal ideal ring. The ideal is generated by the unique monic element in C of minimum degree, the generator polynomial g.[1] This must be a divisor of  . It follows that every cyclic code is a polynomial code. If the generator polynomial g has degree d then the rank of the code C is  .

The idempotent of C is a codeword e such that e2 = e (that is, e is an idempotent element of C) and e is an identity for the code, that is e c = c for every codeword c. If n and q are coprime such a word always exists and is unique;[2] it is a generator of the code.

An irreducible code is a cyclic code in which the code, as an ideal is irreducible, i.e. is minimal in R, so that its check polynomial is an irreducible polynomial.

ExamplesEdit

For example, if A=  and n=3, the set of codewords contained in cyclic code generated by (1,1,0) is precisely

 .

It corresponds to the ideal in   generated by  .

The polynomial   is irreducible in the polynomial ring, and hence the code is an irreducible code.

The idempotent of this code is the polynomial  , corresponding to the codeword (0,1,1).

Trivial examplesEdit

Trivial examples of cyclic codes are An itself and the code containing only the zero codeword. These correspond to generators 1 and   respectively: these two polynomials must always be factors of  .

Over GF(2) the parity bit code, consisting of all words of even weight, corresponds to generator  . Again over GF(2) this must always be a factor of  .

Quasi-cyclic codes and shortened codesEdit

Before delving into the details of cyclic codes first we will discuss quasi-cyclic and shortened codes which are closely related to the cyclic codes and they all can be converted into each other.

DefinitionEdit

Quasi-cyclic codes:[citation needed]

An   quasi-cyclic code is a linear block code such that, for some   which is coprime to  , the polynomial   is a codeword polynomial whenever   is a codeword polynomial.

Here, codeword polynomial is an element of a linear code whose code words are polynomials that are divisible by a polynomial of shorter length called the generator polynomial. Every codeword polynomial can be expressed in the form  , where   is the generator polynomial. Any codeword   of a cyclic code   can be associated with a codeword polynomial, namely,  . A quasi-cyclic code with   equal to   is a cyclic code.

DefinitionEdit

Shortened codes:

An   linear code is called a proper shortened cyclic code if it can be obtained by deleting   positions from an   cyclic code.

In shortened codes information symbols are deleted to obtain a desired blocklength smaller than the design blocklength. The missing information symbols are usually imagined to be at the beginning of the codeword and are considered to be 0. Therefore,    is fixed, and then   is decreased which eventually decreases  . It is not necessary to delete the starting symbols. Depending on the application sometimes consecutive positions are considered as 0 and are deleted.

All the symbols which are dropped need not be transmitted and at the receiving end can be reinserted. To convert   cyclic code to   shortened code, set   symbols to zero and drop them from each codeword. Any cyclic code can be converted to quasi-cyclic codes by dropping every  th symbol where   is a factor of  . If the dropped symbols are not check symbols then this cyclic code is also a shortened code.

Cyclic codes for correcting errorsEdit

Now, we will begin the discussion of cyclic codes explicitly with error detection and correction. Cyclic codes can be used to correct errors, like Hamming codes as a cyclic codes can be used for correcting single error. Likewise, they are also used to correct double errors and burst errors. All types of error corrections are covered briefly in the further subsections.

The (7,4) Hamming code has a generator polynomial  . This polynomial has a zero in Galois extension field   at the primitive element  , and all codewords satisfy  . Cyclic codes can also be used to correct double errors over the field  . Blocklength will be   equal to   and primitive elements   and   as zeros in the   because we are considering the case of two errors here, so each will represent one error.

The received word is a polynomial of degree   given as  

where   can have at most two nonzero coefficients corresponding to 2 errors.

We define the Syndrome Polynomial,   as the remainder of polynomial   when divided by the generator polynomial   i.e.

  as  .

For correcting two errorsEdit

Let the field elements   and   be the two error location numbers. If only one error occurs then   is equal to zero and if none occurs both are zero.

Let   and  .

These field elements are called "syndromes". Now because   is zero at primitive elements   and  , so we can write   and  . If say two errors occur, then

  and  .

And these two can be considered as two pair of equations in   with two unknowns and hence we can write

  and  .

Hence if the two pair of nonlinear equations can be solved cyclic codes can used to correct two errors.

Hamming codeEdit

The Hamming(7,4) code may be written as a cyclic code over GF(2) with generator  . In fact, any binary Hamming code of the form Ham(r, 2) is equivalent to a cyclic code,[3] and any Hamming code of the form Ham(r,q) with r and q-1 relatively prime is also equivalent to a cyclic code.[4] Given a Hamming code of the form Ham(r,2) with  , the set of even codewords forms a cyclic  -code.[5]

Hamming code for correcting single errorsEdit

A code whose minimum distance is at least 3, have a check matrix all of whose columns are distinct and non zero. If a check matrix for a binary code has   rows, then each column is an  -bit binary number. There are   possible columns. Therefore if a check matrix of a binary code with   at least 3 has   rows, then it can only have   columns, not more than that. This defines a   code, called Hamming code.

It is easy to define Hamming codes for large alphabets of size  . We need to define one   matrix with linearly independent columns. For any word of size   there will be columns who are multiples of each other. So, to get linear independence all non zero  -tuples with one as a top most non zero element will be chosen as columns. Then two columns will never be linearly dependent because three columns could be linearly dependent with the minimum distance of the code as 3.

So, there are   nonzero columns with one as top most non zero element. Therefore, Hamming code is a   code.

Now, for cyclic codes, Let   be primitive element in  , and let  . Then   and thus   is a zero of the polynomial   and is a generator polynomial for the cyclic code of block length  .

But for  ,  . And the received word is a polynomial of degree   given as

 

where,   or   where   represents the error locations.

But we can also use   as an element of   to index error location. Because  , we have   and all powers of   from   to   are distinct. Therefore we can easily determine error location   from   unless   which represents no error. So, hamming code is a single error correcting code over   with   and  .

Cyclic codes for correcting burst errorsEdit

From Hamming distance concept, a code with minimum distance   can correct any   errors. But in many channels error pattern is not very arbitrary, it occurs within very short segment of the message. Such kind of errors are called burst errors. So, for correcting such errors we will get a more efficient code of higher rate because of the less constraints. Cyclic codes are used for correcting burst error. In fact, cyclic codes can also correct cyclic burst errors along with burst errors. Cyclic burst errors are defined as

A cyclic burst of length   is a vector whose nonzero components are among   (cyclically) consecutive components, the first and the last of which are nonzero.

In polynomial form cyclic burst of length   can be described as   with   as a polynomial of degree   with nonzero coefficient  . Here   defines the pattern and   defines the starting point of error. Length of the pattern is given by deg . The syndrome polynomial is unique for each pattern and is given by

 

A linear block code that corrects all burst errors of length   or less must have at least   check symbols. Proof: Because any linear code that can correct burst pattern of length   or less cannot have a burst of length   or less as a codeword because if it did then a burst of length   could change the codeword to burst pattern of length  , which also could be obtained by making a burst error of length   in all zero codeword. Now, any two vectors that are non zero in the first   components must be from different co-sets of an array to avoid their difference being a codeword of bursts of length  . Therefore number of such co-sets are equal to number of such vectors which are  . Hence at least   co-sets and hence at least   check symbol.

This property is also known as Rieger bound and it is similar to the singleton bound for random error correcting.

Fire codes as cyclic boundsEdit

In 1959, Philip Fire[6] presented a construction of cyclic codes generated by a product of a binomial and a primitive polynomial. The binomial has the form   for some positive odd integer  .[7] Fire code is a cyclic burst error correcting code over   with the generator polynomial

 

where   is a prime polynomial with degree   not smaller than   and   does not divide  . Block length of the fire code is the smallest integer   such that   divides  .

A fire code can correct all burst errors of length t or less if no two bursts   and   appear in the same co-set. This can be proved by contradiction. Suppose there are two distinct nonzero bursts   and   of length   or less and are in the same co-set of the code. So, their difference is a codeword. As the difference is a multiple of   it is also a multiple of  . Therefore,

 .

This shows that   is a multiple of  , So

 

for some  . Now, as   is less than   and   is less than   so   is a codeword. Therefore,

 .

Since   degree is less than degree of  ,  cannot divide  . If   is not zero, then   also cannot divide   as   is less than   and by definition of  ,   divides   for no   smaller than  . Therefore   and   equals to zero. That means both that both the bursts are same, contrary to assumption.

Fire codes are the best single burst correcting codes with high rate and they are constructed analytically. They are of very high rate and when   and   are equal, redundancy is least and is equal to  . By using multiple fire codes longer burst errors can also be corrected.

For error detection cyclic codes are widely used and are called   cyclic redundancy codes.

Cyclic codes on Fourier transformEdit

Applications of Fourier transform are widespread in signal processing. But their applications are not limited to the complex fields only; Fourier transforms also exist in the Galois field  . Cyclic codes using Fourier transform can be described in a setting closer to the signal processing.

Fourier transform over finite fieldsEdit

Fourier transform over finite fields

The discrete Fourier transform of vector   is given by a vector   where,

  =   where,

 

where exp( ) is an  th root of unity. Similarly in the finite field  th root of unity is element   of order  . Therefore

If   is a vector over  , and   be an element of   of order  , then Fourier transform of the vector   is the vector   and components are given by

  =   where,

 

Here   is time index,   is frequency and   is the spectrum. One important difference between Fourier transform in complex field and Galois field is that complex field   exists for every value of   while in Galois field   exists only if   divides  . In case of extension fields, there will be a Fourier transform in the extension field   if   divides   for some  . In Galois field time domain vector   is over the field   but the spectrum   may be over the extension field  .

Spectral description of cyclic codesEdit

Any codeword of cyclic code of blocklength   can be represented by a polynomial   of degree at most  . Its encoder can be written as  . Therefore in frequency domain encoder can be written as  . Here codeword spectrum   has a value in   but all the components in the time domain are from  . As the data spectrum   is arbitrary, the role of   is to specify those   where   will be zero.

Thus, cyclic codes can also be defined as

Given a set of spectral indices,  , whose elements are called check frequencies, the cyclic code   is the set of words over   whose spectrum is zero in the components indexed by  . Any such spectrum   will have components of the form  .

So, cyclic codes are vectors in the field   and the spectrum given by its inverse fourier transform is over the field   and are constrained to be zero at certain components. But every spectrum in the field   and zero at certain components may not have inverse transforms with components in the field  . Such spectrum can not be used as cyclic codes.

Following are the few bounds on the spectrum of cyclic codes.

BCH boundEdit

If   be a factor of   for some  . The only vector in   of weight   or less that has   consecutive components of its spectrum equal to zero is all-zero vector.

Hartmann-Tzeng boundEdit

If   be a factor of   for some  , and   an integer that is coprime with  . The only vector   in   of weight   or less whose spectral components   equal zero for  , where   and  , is the all zero vector.

Roos boundEdit

If   be a factor of   for some   and  . The only vector in   of weight   or less whose spectral components   equal to zero for  , where   and   takes at least   values in the range  , is the all-zero vector.

Quadratic residue codesEdit

When the prime   is a quadratic residue modulo the prime   there is a quadratic residue code which is a cyclic code of length  , dimension   and minimum weight at least   over  .

GeneralizationsEdit

A constacyclic code is a linear code with the property that for some constant λ if (c1,c2,...,cn) is a codeword then so is (λcn,c1,...,cn-1). A negacyclic code is a constacyclic code with λ=-1.[8] A quasi-cyclic code has the property that for some s, any cyclic shift of a codeword by s places is again a codeword.[9] A double circulant code is a quasi-cyclic code of even length with s=2.[9]

See alsoEdit

NotesEdit

  1. ^ Van Lint 1998, p. 76
  2. ^ Van Lint 1998, p. 80
  3. ^ Hill 1988, pp. 159-160
  4. ^ Blahut 1983, Theorem 5.5.1
  5. ^ Hill 1988, pp. 162-163
  6. ^ P. Fire, E, P. (1959). A class of multiple-error-correcting binary codes for non-independent errors. Sylvania Reconnaissance Systems Laboratory, Mountain View, CA, Rept. RSL-E-2, 1959.
  7. ^ Wei Zhou, Shu Lin, Khaled Abdel-Ghaffar. Burst or random error correction based on Fire and BCH codes. ITA 2014: 1-5 2013.
  8. ^ Van Lint 1998, p. 75
  9. ^ a b MacWilliams & Sloane 1977, p. 506

ReferencesEdit

  • Blahut, Richard E. (2003), Algebraic Codes for Data Transmission (2nd ed.), Cambridge University Press, ISBN 0-521-55374-1
  • Hill, Raymond (1988), A First Course In Coding Theory, Oxford University Press, ISBN 0-19-853803-0
  • MacWilliams, F. J.; Sloane, N. J. A. (1977), The Theory of Error-Correcting Codes, New York: North-Holland Publishing, ISBN 0-444-85011-2
  • Van Lint, J. H. (1998), Introduction to Coding Theory, Graduate Texts in Mathematics 86 (3rd ed.), Springer Verlag, ISBN 3-540-64133-5

Further readingEdit

External linksEdit

This article incorporates material from cyclic code on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.