Cyclic Codes Of Odd Length Research Articles

Some results on cyclic codes over the ring $R_{2,m}$

Let Rk;m be the ring F2m(u1;u2; � � � ;uk)= ⟨ u 2;uiuj ujui ⟩ . In this paper, cyclic codes of arbitrary length n over the ring R2;m are completely characterized in terms of unique generators and a way for determination of these generators is investigated. A F2m -basis for these codes is also derived from this representation. Moreover, it is proven that there exists a one-to-one correspondence between cyclic codes of length 2n, n odd, over the ring Rk 1;m and cyclic codes of length n over the ring Rk;m. By determining the complete structure of cyclic codes of length 2 over R2;m, a mass formula for the number of these codes is given. Using this and the mentioned correspondence, the number of ideals of the rings R2;m and R3;m is determined. As a corollary, the number of cyclic codes of odd length n over the rings R2;m and R3;m is obtained.

Additive semisimple multivariable codes over $${\mathbb{F}_4}$$

The structure of additive multivariable codes over $${\mathbb{F}_4}$$ (the Galois field with 4 elements) is presented. The semisimple case (i.e., when the defining polynomials of the code have no repeated roots) is specifically addressed. These codes extend in a natural way the abelian codes, of which additive cyclic codes of odd length are a particular case. Duality of these codes is also studied.

Some Results on Cyclic Codes Over ${F}_{2}+v{F}_{2}$

In this paper, we investigate the structure and properties of cyclic codes over the ring <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vF</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . We first study the relationship between cyclic codes over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vF</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent generators for cyclic codes of odd length and determine the number of cyclic codes for a given length <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">vF</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> .

Additive cyclic codes over $\mathbb F_4$

In this paper we find a canonical form decomposition for additive cyclic codes of odd length over $\mathbb F_4$. This decomposition is used to count the number of such codes. We also reprove that each code is the $\mathbb F_2$-span of at most two codewords and their cyclic shifts, a fact first proved in [2]. A count is given for the number of codes that are the $\mathbb F_2$-span of one codeword and its cyclic shifts. We can examine this decomposition to see precisely when the code is self-orthogonal or self-dual under the trace inner product. Using this, a count is presented for the number of self-orthogonal and self-dual additive cyclic codes of odd length. We also provide a count of the additive cyclic and additive cyclic self-orthogonal codes as a function of their $\mathbb F_2$-dimension.

Construction of cyclic codes over [formula omitted] for DNA computing

In this paper, we develop the theory for constructing linear and additive cyclic codes of odd length over GF ( 4 ) that are suitable for DNA computing. We call this class of codes reversible complement cyclic codes. We use this theory to study all such codes of lengths 7 , 9 , 11 and 13 . We list the codes that have the largest number of codewords for a given minimum Hamming distance. We show that some of these codes have more codewords than previously known codes with the same minimum Hamming distance.

Cyclic codes of length 2 e over Z4

Cyclic codes of odd length over Z 4 have been studied by many authors. But what is the form of cylic codes of even length? The structure of cyclic codes of length n=2 e , for any positive integer e is considered. We show that any cyclic code is an ideal in the ring R n = Z 4[ x]/〈 x n −1〉. We show that the ring R n is a local ring but not a principal ideal ring. Also, we find the set of generators for cyclic codes. Examples of cyclic codes of such length are given.

Cyclic Codes of Length 2e over Z4

Abstract Cyclic codes of odd length over Z4 have been studies by many outhers. But what is the form of cyclic codes of even length? The structure of cyclic codes of length 2e, for any positive integer n, is considered. We will show that any cyclic code is an ideal in the ring Rn = Z4[x]/ 〈xn − 1〉. We will show that the ring Rn is a local ring but not a principal ideal ring. Also, we will find the set of generators for cyclic codes. Examples of cyclic codes will be given.

Decoding of cyclic codes over F/sub 2/+uF/sub 2/

We give a simple decoding algorithm to decode linear cyclic codes of odd length over the ring R=F/sub 2/+uF/sub 2/={0,1,u,u~=u+1}, where u/sup 2/=0. A spectral representation of the cyclic codes over R is given and a BCH-like bound is given for the Lee distance of the codes. The ring R shares many properties of Z/sub 4/ and F/sub 4/ and admits a linear "Gray map".

On the generalized Hamming weights of cyclic codes

We prove several results on the generalized Hamming weights (GHW's) of linear codes, particularly for cyclic codes. Based on these and previously known results, we give some efficient algorithms for computing GHW hierarchy of cyclic codes. We give complete weight hierarchy for each of the binary cyclic codes of odd lengths /spl les/31. A table of second and third GHW's of binary cyclic codes of odd lengths /spl les/57 is also presented. We have also computed the second GHW of all binary cyclic codes of length 63 and the third GHW of one code from each dimension.

On binary cyclic codes of odd lengths from 101 to 127

All binary cyclic codes of odd lengths are checked from 101 to 127 to find codes which are better than those in a table by T. Verhoeff (1989). There are five such cases, namely, (117, 36, 32), (117, 37, 29), (117, 42, 26), (117, 49, 24), and (127, 36 35) cyclic codes. According to Verhoeff's table the previously known ranges of the highest minimum-distance were 28-40, 28-40, 25-37, 22-32, and 32-46, respectively. Applying constructions X and Y1, (120, 37, 32) and (108, 28, 32) codes were found. Moreover, the highest minimum-distances that cyclic codes of length 127 can attain are determined. >

Computer search for binary cyclic UEP codes of odd length up to 65

Exhaustive computation by a computer was used to find the unequal error protection capabilities of all binary cyclic codes of odd length up to 65 that have minimum distances of at least 3. For those codes for which upper bounds can only be computed on their unequal error protection capabilities, an analytic method developed by V.N. Dynkin and V.A. Togonidze (1976) is used to show that the upper bounds meet the exact unequal error protection capabilities.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The minimum distance of all binary cyclic codes of odd lengths from 69 to 99

A computer search has been made to determine the true minimum distance d for all binary cyclic codes having odd lengths n in the range 69\leq n\leq 99 . Using an algorithm originally developed by C. L. Chen, the generator matrix G of each (n,k) binary cyclic code was put in systematic form. All possible codewords obtained from sums of i rows of G , for i= 1,2, \cdots,\upsilon , were examined, and the minimum distance d_{\upsilon} of this set was recorded. Then d=d_{\upsilon} whenever \upsilon >\{(d_{\upsilon}-l)k/n\}-l . Known equivalences among cyclic codes were taken into account, and only one code from each equivalence class was listed. Let g(x) divide x^{n}-1 , where x - 1 is not a factor of g(x) . Then the minimum distances of the codes generated by g(x),(x - l)g(x) and their duals are listed together. For each such set of codes, the value of \upsilon for which a codeword of minimum weight first appeared is listed. The codes found were compared with the list of best codes tabulated by Sloane [5]. Many good cyclic codes have been found. Among the best (n,k,d) cyclic codes found are the following: (73, 27, 20), (73, 36, 16), (85, 12, 34), (85, 20, 28), (87, 31, 22), (89, 56, 11), (91, 51, 14), (93, 20, 32), (93, 23, 29), (93, 31, 24), (93, 33, 22).