Cyclic, LCD, and Self-Dual Codes over the Non-Frobenius Ring GR(p2,m)[u]/⟨u2,pu⟩

On rank and MDR cyclic and negacyclic codes of length [formula omitted] over [formula omitted

Lee weights of cyclic self-dual codes over Galois rings of characteristic p2

This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances.

Let $R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle $. Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n over R and provide a clear formula to count the number of these self-dual cyclic codes. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes of length 30 over $\mathbb {Z}_{4}$ and obtain 4-quasi-cyclic and formally self-dual binary linear [60,30,12] codes derived from cyclic codes of length 15 over $\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle $.

Cyclic linear codes of block length n over a finite field F/sub q/ are linear subspaces of F/sub q//sup n/ that are invariant under a cyclic shift of their coordinates. A family of codes is good if all the codes in the family have constant rate and constant normalized distance (distance divided by block length). It is a long-standing open problem whether there exists a good family of cyclic linear codes. A code C is r-testable if there exists a randomized algorithm which, given a word x/spl isin//sub q//sup n/, adaptively selects r positions, checks the entries of x in the selected positions, and makes a decision (accept or reject x) based on the positions selected and the numbers found, such that 1) if x/spl isin/C then x is surely accepted; ii) if dist(x,C) /spl ges/ /spl epsi/n then x is probably rejected. (dist refers to Hamming distance.) A family of codes is locally testable if all members of the family are r-testable for some constant r. This concept arose from holographic proofs/PCP's. Recently it was asked whether there exist good, locally testable families of codes. In this paper the intersection of the two questions stated is addressed. Theorem. There are no good, locally testable families of cyclic codes over any (fixed) finite field. In fact the result is stronger in that it replaces condition ii) of local testability by the condition ii') if dist (x,C) /spl ges/ /spl epsi/n then x has a positive chance of being rejected. The proof involves methods from Galois theory, cyclotomy, and diophantine approximation.

In this paper, we study the cyclic self-orthogonal codes over a finite commutative chain ring [Formula: see text], where [Formula: see text] is a prime number. A generating polynomial of cyclic self-orthogonal codes over [Formula: see text] is obtained. We also provide a necessary and sufficient condition for the existence of nontrivial self-orthogonal codes over [Formula: see text]. Finally, we determine the number of the above codes with length [Formula: see text] over [Formula: see text] for any [Formula: see text]. The results are given by Zhe-Xian Wan on cyclic codes over Galois rings in [Z. Wan, Cyclic codes over Galois rings, Algebra Colloq. 6 (1999) 291–304] are extended and strengthened to cyclic self-orthogonal codes over [Formula: see text].

We investigate negacyclic codes over the Galois ring GR(2 a ,m) of length N = 2 k n, where n is odd and k ⩾ 0. We first determine the structure of u-constacyclic codes of length n over the finite chain ring $GR(2^a ,m)[u]/\langle u^{2^k } + 1\rangle $ . Then using a ring isomorphism we obtain the structure of negacyclic codes over GR(2 a ,m) of length N = 2 k n (n odd) and explore the existence of self-dual negacyclic codes over GR(2 a ,m). A bound for the homogeneous distance of such negacyclic codes is also given.

We classify all the cyclic self-dual codes of length $$p^k$$ over the finite chain ring $$\mathcal R:=\mathbb Z_p[u]/\langle u^3 \rangle $$ , which is not a Galois ring, where p is a prime number and k is a positive integer. First, we find all the dual codes of cyclic codes over $${\mathcal R}$$ of length $$p^k$$ for every prime p. We then prove that if a cyclic code over $${\mathcal R}$$ of length $$p^k$$ is self-dual, then p should be equal to 2. Furthermore, we completely determine the generators of all the cyclic self-dual codes over $$\mathbb Z_2[u]/\langle u^3 \rangle $$ of length $$2^k$$ . Finally, we obtain a mass formula for counting cyclic self-dual codes over $$\mathbb Z_2[u]/\langle u^3 \rangle $$ of length $$2^k$$ .

A large class of cyclic and shortened cyclic binary codes for multiple error correction

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An alternative algorithm for factorizing $x^n-\lambda $ over ${\mathbb{F}_{{q^2}}}$ is given, where $λ$ is a unit in ${\mathbb{F}_{{q^2}}}$. Based on this factorization, the dimensions of the Hermitian hulls of $\lambda $-constacyclic codes of length $n$ over ${\mathbb{F}_{{q^2}}}$ are determined. The characterization and enumeration of constacyclic Hermitian self-dual (resp., complementary dual) codes of length $n$ over ${\mathbb{F}_{{q^2}}}$ are given through their Hermitian hulls. Subsequently, a new family of MDS constacyclic Hermitian self-dual codes over ${\mathbb{F}_{{q^2}}}$ is introduced.As a generalization of constacyclic codes, quasi-twisted Hermitian self-dual codes are studied. Using the factorization of $x^n-\lambda $ and the Chinese Remainder Theorem, quasi-twisted codes can be viewed as a product of linear codes of shorter length over some extension fields of ${\mathbb{F}_{{q^2}}}$. Necessary and sufficient conditions for quasi-twisted codes to be Hermitian self-dual are given. The enumeration of such self-dual codes is determined as well.

Constacyclic codes form an interesting family of error-correcting codes due to their rich algebraic structure, and are generalizations of cyclic and negacyclic codes. In this paper, we classify repeated-root constacyclic codes of length l t p s over the finite field F p m $\mathbb {F}_{p^{m}}$ containing p m elements, where l ? 1(mod 2), p are distinct primes and t, s, m are positive integers. Based upon this classification, we explicitly determine the algebraic structure of all repeated-root constacyclic codes of length l t p s over F p m $\mathbb {F}_{p^{m}}$ and their dual codes in terms of generator polynomials. We also observe that self-dual cyclic (negacyclic) codes of length l t p s over F p m $\mathbb {F}_{p^{m}}$ exist only when p = 2 and list all self-dual cyclic (negacyclic) codes of length l t 2 s over F 2 m $\mathbb {F}_{2^{m}}$ . We also determine all self-orthogonal cyclic and negacyclic codes of length l t p s over F p m $\mathbb {F}_{p^{m}}$ . To illustrate our results, we determine all constacyclic codes of length 175 over F 5 $\mathbb {F}_{5}$ and all constacyclic codes of lengths 147 and 3087 over F 7 $\mathbb {F}_{7}$ .

Cyclic codes having length ptq have been studied using the primitive binary idempotent generators when ( ) 2 (2) = pt pt O f and (2) = 1 qO q − with ( ( ) ) 2 , 1 = 1. f pt q − Using the expressions of these idempotent generators and the theory of cyclotomy, expressions of idempotent generators of ℤ4 – cyclic codes are obtained. In this case, all the self-dual codes and LCD codes of length ptq over ℤ4 are defined in terms of the idempotent generators. Clearly, these self-dual cyclic codes of length ptq are Type-I codes. Also, the permutational equivalence of these cyclic codes has been discussed.

It has been previously shown [5], that a binary linear cyclic code of length 2n (n odd) can be obtained from two binary linear cyclic codes of length n by the well known |u|u+v| construction. It is easy to show that the same construction can as well be obtained as the image under the Gray map of a cyclic (not necessarily linear) code of length n over ZZ4. We shall show that the set of linear cyclic codes over ZZ4, whose images under the Gray map agree with the |u|u+v| construction, are the same family of codes whose images correspond to binary linear cyclic codes of length 2n under the Nechaev-Gray map introduced in [11]. Since the number of linear cyclic codes of length n over ZZ4 is equal to the number of binary linear cyclic codes of length 2n, we use this result to characterize the set of nonlinear cyclic codes of length n over ZZ4, whose images, under the Nechaev-Gray map, are binary linear cyclic codes of length 2n. As a byproduct, we introduce a new product for binary polinomials, and, by means of this product, we obtain a new way to express codewords that belong to a linear cyclic code over ZZ4 whose Nechaev-Gray image is a binary linear 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.

Cyclic codes over [formula omitted] and applications to DNA codes