Binary Cyclic Codes Of Length Research Articles

Cyclic Codes over a Non-Commutative Non-Unital Ring

In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as E=⟨a,b∣2a=2b=0,a2=a,b2=b,ab=a,ba=b⟩. This is the first time that cyclic codes over the ring E are studied. Each cyclic code of length n over E is identified uniquely by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, left self-dual, right self-dual, and linear complementary dual (LCD) cyclic codes over E. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over E and quasi-cyclic codes of length 2n over F2 is studied. Motivated by DNA computing, conditions for reversibility and invariance under complementation are derived.

Cyclic Codes over a Non-Local Non-Unital Ring

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as H=⟨a,b∣2a=2b=0,a2=0,b2=b,ab=ba=0⟩. This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length n. We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes H. We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length 2n over F2 is studied.

Additive and linear conjucyclic codes over $ {\mathbb{F}}_4 $

<p style='text-indent:20px;'>Conjucyclic codes were first introduced by Calderbank, Rains, Shor and Sloane in [<xref ref-type="bibr" rid="b1">1</xref>] because of their applications in quantum error-correction. In this paper, we study linear and additive conjucyclic codes over the finite field <inline-formula><tex-math id="M2">\begin{document}$ {\mathbb{F}}_{4} $\end{document}</tex-math></inline-formula> and produce a duality for which the orthogonal, with respect to that duality, of conjucyclic codes is conjucyclic. Moreover, we show that this is not the case for other standard dualities. We show that additive conjucyclic codes are the only non-trivial conjucyclic codes over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb{F}}_{4} $\end{document}</tex-math></inline-formula> and we use a linear algebraic approach to classify these codes. We will also show that additive conjucyclic codes of length <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M5">\begin{document}$ {\mathbb{F}}_{4} $\end{document}</tex-math></inline-formula> are isomorphic to binary cyclic codes of length <inline-formula><tex-math id="M6">\begin{document}$ 2n. $\end{document}</tex-math></inline-formula></p>

Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer t ⩾ 3 is said to be exceptional if f ( x ) = x t is APN (Almost Perfect Nonlinear) over F 2 n for infinitely many values of n. Equivalently, t is exceptional if the binary cyclic code of length 2 n − 1 with two zeros ω , ω t has minimum distance 5 for infinitely many values of n. The conjecture we prove states that every exceptional number has the form 2 i + 1 or 4 i − 2 i + 1 .

The weight distributions of cyclic codes with two zeros and zeta functions

We consider the weight distribution of the binary cyclic code of length 2 n − 1 with two zeros α a , α b . Our proof gives information in terms of the zeta function of an associated variety. We carry out an explicit determination of the weight distribution in two cases, for the cyclic codes with zeros α 3 , α 5 and α , α 11 . These are the smallest cases of two infinite families where finding the weight distribution is an open problem. Finally, an interesting application of our methods is that we can prove that these two codes have the same weight distribution for all odd n .

New linear codes derived from binary cyclic codes of length 151

The minimum distance of all binary cyclic codes of length 151 is determined. Almost all of these cyclic codes have the same parameters as the best linear codes given in Brouwer's database. A nested chain of linear codes is derived from these cyclic codes and some new binary codes are obtained by applying Constructions X and XX to pairs of codes in this chain. Good candidates for nested codes can also be obtained by enlarging the cyclic codes of high minimum distance. In total, there are 39 new binary linear codes that have a minimum distance higher than codes previously considered to be the best linear codes.

QUANTUM CODES CONSTRUCTED FROM BINARY CYCLIC CODES

In this paper, we use 2-cyclotomic cosets of modulo n and generator polynomials to describe binary cyclic codes of length N=2αn with n odd. We discuss the conditions under which two cyclic codes [Formula: see text] and [Formula: see text] can be used to construct quantum codes by CSS construction or Steane's construction. Using the results of Chen, Promhouse and Tavares, and Castagnoli et al., we study the quantum codes that can be constructed from binary cyclic codes of length N=2αn with n odd and n≤99, and α≤2. We find that except the quantum codes constructed by Steane, there are also some very interesting quantum codes constructed from repeated-root cyclic codes, and some of the quantum codes constructed by Steane can be improved.

On three weights in cyclic codes with two zeros

We show that the dual code of the binary cyclic code of length 2 m −1 with two zeros α, α d cannot have three weights in the case that m is even and d≡0 ( mod 3) . The proof involves the partial calculation of a coset weight distribution.

On binary cyclic codes with codewords of weight three and binary sequences with the trinomial property

Golomb and Gong (1997 and 1999) considered binary sequences with the trinomial property. In this correspondence we shall show that the sets of those sequences are (quite trivially) closely connected with binary cyclic codes with codewords of weight three. This approach gives us another way to deal with trinomial property problems. After disproving one conjecture formulated by Golomb and Gong, we exhibit an infinite class of sequences which do not have the trinomial property, corresponding to binary cyclic codes of length 2/sup m/-1 with minimum distance exactly four.

Generalized cyclotomic codes of length p/sub 1//sup e(1)/...p/sub t//sup e(t)/

We first introduce a generalized cyclotomy of order 2 with respect to p/sub 1//sup e(1)/...p/sub t//sup e(t)/. We then present two classes of new error-correcting binary cyclic codes of length p/sub 1//sup e(1)/...p/sub t//sup e(t)/ based on this generalized cyclotomy. We prove either a square-root bound or a similar bound on the minimum odd weight of those codes with length n=p/sub 1//sup e(1)/ p/sub 2//sup e(2)/.

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.

Generation of matrices for determining minimum distance and decoding of cyclic codes

A simple method based on Newton's identities and their extensions is presented for determining the actual minimum distance of cyclic codes. More significantly, it is shown that this method also provides a mechanism for generating the type of syndrome matrices needed by Feng and Tzeng's (see ibid., vol.40, p.1364-1374, Sept. 1994) new procedure for decoding cyclic and BCH codes up to their actual minimum distance. Two procedures for generating such matrices are given. With these procedures, we have generated syndrome matrices having only one class of conjugate syndromes on the minor diagonal for all binary cyclic codes of length n<63 and many codes of length 63/spl les/n/spl les/99. A listing of such syndrome matrices for selected codes of length n<63 is included. An interesting connection of the method presented to the shifting technique of van Lint (1986) and Wilson is also noted.

Decoding short binary cyclic codes via covering polynomials

AbstractThe paper concerns the covering polynomial method of decoding cyclic codes. This is an easily‐implementable, robust, and low‐complexity decoding algorithm. The determination of the exact complexity of the algorithm is a difficult open problem. In this paper several methods for constructing minimal sets of covering polynomials are proposed for the case R &lt; 2/τ, where τ is the number of errors to be corrected. Complete closed form specifications of the minimal set are given for the case where τ is even, and a computationally efficient procedure presented for general τ is shown to be optimal in many cases (and conjectured to be globally optimal). These results are sufficient to determine the exact minimal number of covering monomials for all binary cyclic codes of length up to 127 with τ &lt; 9. Closed form specifications are available for the covering polynomial set for all such codes.

Studying the locator polynomials of minimum weight codewords of BCH codes

Primitive binary cyclic codes of length n=2/sup m/ are considered. A BCH code with designed distance delta is denoted B(n, delta ). A BCH code is always a narrow-sense BCH code. A codeword is identified with its locator polynomial, whose coefficients are the symmetric functions of the locators. The definition of the code by its zeros-set involves some properties for the power sums of the locators. Moreover, the symmetric functions and the power sums of the locators are related to Newton's identities. An algebraic point of view is presented in order to prove or disprove the existence of words of a given weight in a code. The principal result is the true minimum distance of some BCH codes of length 255 and 511. which were not known. The minimum weight codewords of the codes B(n2/sup h/-1) are studied. It is proved that the set of the minimum weight codewords of the BCH code B(n,2/sup m-2/-1) equals the set of the minimum weight codewords of the punctured Reed-Muller code of length n and order 2, for any m.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Covering radius computations for binary cyclic codes

We compute the covering radius of each binary cyclic code of length $\leq 64$ (for both even and odd lengths) and redundancy $\leq 28$. We also compute the covering radii of their punctured codes and shortened codes. Thus we give exact covering radii of over six thousand codes. For each of these codes (except for certain composite codes), we also determine the number of cosets of each weight less than or equal to the covering radius. These results are used to compute the minimum distances of the above cyclic codes. We use the covering radii of shortened codes and other criteria for normality to show that all but eight of the cyclic codes for which we determine the covering radius are normal. For all but seven of these normal codes, we determine the norm using some old results and some new results proved here. We observe that many cyclic codes are among the best covering codes discovered so far, and some of them lead to improvements on the previously published bounds on $t[n,k]$, the smallest covering radius of any binary linear [

Covering Radius Computations for Binary Cyclic Codes

We compute the covering radius of each binary cyclic code of length $\leq 64$ (for both even and odd lengths) and redundancy $\leq 28$. We also compute the covering radii of their punctured codes and shortened codes. Thus we give exact covering radii of over six thousand codes. For each of these codes (except for certain composite codes), we also determine the number of cosets of each weight less than or equal to the covering radius. These results are used to compute the minimum distances of the above cyclic codes. We use the covering radii of shortened codes and other criteria for normality to show that all but eight of the cyclic codes for which we determine the covering radius are normal. For all but seven of these normal codes, we determine the norm using some old results and some new results proved here. We observe that many cyclic codes are among the best covering codes discovered so far, and some of them lead to improvements on the previously published bounds on $t[n,k]$, the smallest covering radius of any binary linear [n, k] code. Among some other applications of our results, we use our table of covering radii and a code augmentation argument to give four improvements on the values of ${d_{\max }}(n,k)$, where ${d_{\max }}(n,k)$ is the largest minimum distance of any binary [n, k] code. These results show that the covering radius is intimately connected with the other three parameters of a linear code, n, k, and d. We also give a complete classification (up to isomorphism) of cyclic self-dual codes of lengths 42, 56, and 60. The computations were carried out mainly on concurrent machines (hypercubes and Connection Machines); we give a description of our algorithm.

A transform theory for a class of group-invariant codes

A binary cyclic code of length of n can be defined by a set of parity-check equations that is invariant under the additive cyclic group generated by A:i to i+1 modulo n. A class of quasicyclic codes that have parity-check equations invariant under the group generated by a subgroup of A and a subgroup of the multiplicative group M:i to 2/sup k/i modulo n is studied in detail. The Fourier transform transforms the action of the additive group, and a linearized polynomial transform transforms the action of the multiplicative group. The form of the transformed code equations permits a BCH-like bound on the minimum distance of the code. The techniques are illustrated by the construction of several codes that are decodable by the Berlekamp-Massey algorithm and that have parameters that approach or surpass those of the best codes known. >

Binary cyclic codes generated by mira7 (Corresp.)

We show that binary cyclic codes of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n = 2^{m} - 1</tex> with generator <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(x)=m_{1}(x)m_{7}(x)</tex> have minimum distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&lt; 5</tex> unless <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m = 5</tex> (and possibly <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m = 11, 13, or 17)</tex> .

On the minimum distance of cyclic codes

The main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds (i.e., BCH bound, HT bound, Roos bound). This bound is related to a second method for bounding the minimum distance of a cyclic code, which we call shifting. This method can be even stronger than the first one. For all binary cyclic codes of length (with two exceptions), we show that our methods yield the true minimum distance. The two exceptions at the end of our list are a code and its even-weight subcode. We treat several examples of cyclic codes of length \geq 63 .

On a necessary condition for L-step orthogonalization of linear codes and its applications

By generalizing a known result for double-error-correcting binary primitive Bose\3-Chaudhuri\3-Hocquenghem codes, the following statement is proved: A necessary condition for L-step orthogonalization of a t -error-correcting binary linear code C of length n is given by n ⩾ {if123-1} + {if123-2} if its expurgated version C ′ is distinct from C , and n ⩾ {if123-3} if C ′ is identical to C , where {if123-4} is the minimum distance of the dual {if123-5} of the code C ′, and {if123-6} is that of the dual {if123-7} of the code C ′, expurgated from C by a single information digit. From this result, it is found that the following codes cannot be L -step orthogonalized: triple-error-correcting BCH codes with odd m greater than 5 and even m equal to 8 and 10, all the BCH codes with m = 7 except two trivial cases, and most of the binary quadratic residue codes with known minimum Hamming distances. In addition, various codes are analyzed; and where possible, the types of majority-logic decoding are tabulated for these codes, namely, a class of binary cyclic codes of length n ⩽ 43 and all the binary BCH codes of length n = 63.