Repeated-root Cyclic Codes Research Articles

Correction: Determination for minimum symbol-pair and RT weights via torsional degrees of repeated-root cyclic codes

Correction: Determination for minimum symbol-pair and RT weights via torsional degrees of repeated-root cyclic codes

Determination for minimum symbol-pair and RT weights via torsional degrees of repeated-root cyclic codes

Determination for minimum symbol-pair and RT weights via torsional degrees of repeated-root cyclic codes

MDS and AMDS symbol-pair codes constructed from repeated-root cyclic codes

Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against the pair errors in symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. Based on repeated-root cyclic codes, we construct two classes of MDS symbol-pair codes for more general generator polynomials and also give a new class of almost MDS (AMDS) symbol-pair codes with the length lp. In addition, we derive all MDS and AMDS symbol-pair codes with length 3p, when the degree of the generator polynomials is no more than 10. The main results are obtained by determining the solutions of certain equations over finite fields.

AMDS symbol-pair codes from repeated-root cyclic codes

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance is of significance in determining the error-correcting capability of a symbol-pair code. One of the central themes in symbol-pair coding theory is the constructions of symbol-pair codes with the largest possible minimum symbol-pair distance. Maximum distance separable (MDS) and almost maximum distance separable (AMDS) symbol-pair codes are optimal and sub-optimal regarding the Singleton bound, respectively. In this paper, six new classes of AMDS symbol-pair codes are explicitly constructed through repeated-root cyclic codes. Remarkably, one class of such codes has unbounded lengths and the minimum symbol-pair distance of another class can reach 13.

Constructions of MDS symbol-pair codes with minimum distance seven or eight

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance plays a vital role in determining the error-correcting capability and the constructions of symbol-pair codes with largest possible minimum symbol-pair distance is of great importance. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that such codes can acheive the Singleton bound. In this paper, for length 5p, two new classes of MDS symbol-pair codes with minimum symbol-pair distance seven or eight are constructed by utilizing repeated-root cyclic codes over $${\mathbb {F}}_{p}$$ , where p is a prime. In addition, we derive a class of MDS symbol-pair codes with minimum symbol-pair distance seven and length 4p.

MDS symbol-pair codes from repeated-root cyclic codes

Symbol-pair codes are proposed to protect against pair-errors in symbol-pair read channels. The research of symbol-pair codes with the largest possible minimum symbol-pair distance is of great significance since such codes have the best error-correcting capability. A symbol-pair code meeting the Singleton bound is called a maximum distance separable (MDS) symbol-pair code. In this paper, two new classes of MDS symbol-pair codes are proposed by utilizing repeated-root cyclic codes over finite fields with odd characteristic. It should be noted that these codes have minimum symbol-pair distance ten or twelve.

Repeated root cyclic codes over $\mathbb {Z}_{p^{2}}+u\mathbb {Z}_{p^{2}}$ and their Lee distances

Repeated root cyclic codes over $\mathbb {Z}_{p^{2}}+u\mathbb {Z}_{p^{2}}$ and their Lee distances

Quantum Codes from Repeated-Root Cyclic and Negacyclic Codes of Length 4ps Over $\mathbb {F}_{p^{m}}$

Let $\mathbb {F}_{p^{m}}$ be a finite field with pm elements for an odd prime p and a positive integer m. In this paper, we aim to construct non-binary quantum codes from repeated root cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ . To achieve this, first we discuss the structure of all maximum distance separable (MDS) cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ and their Euclidean and Hermitian duals. Further, we establish a necessary and sufficient condition for these codes to contain their duals. Finally, many new and quantum MDS codes are obtained with the help of CSS and Hermitian constructions from dual containing cyclic and negacyclic codes.

The minimum Hamming distances of repeated-root cyclic codes of length $$6p^s$$ and their MDS codes

In this paper, let p be a prime with $$p\ge 7$$ . We determine the weight distributions of cyclic codes of length 6 over $$\mathbb {F}_q$$ and the minimum Hamming distances of all repeated-root cyclic codes of length $$6p^s$$ over $$\mathbb {F}_q$$ , where q is a power of p and s is an integer. Furthermore, we find all maximum distance separable codes of length $$6p^s$$ .

The Hamming distances of repeated-root cyclic codes of length [formula omitted

Due to the wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. In this paper, let p be a prime with p≥7. We determine the Hamming distances of all repeated-root cyclic codes of length 5ps over Fq, where q=pm and both s and m are positive integers. Furthermore, we find all MDS cyclic codes of length 5ps and take quantum synchronizable codes from repeated-root cyclic codes of length 5ps. By comparing the minimum distances of 5ps-length repeated-root cyclic codes to BCH codes of close lengths, we illustrated that quantum synchronizable codes constructed from repeated-root cyclic codes have in general better performance in correcting Pauli errors than those from BCH codes.

On a Family of Quantum Synchronizable Codes Based on the $(\lambda(u + v)|u - v)$ Construction

In this paper, we propose a family of quantum synchronizable codes from repeated-root cyclic codes and constacyclic codes. This family of quantum synchronizable codes are based on $(\lambda (u + v)|u - v)$ construction which is constructed from constacyclic codes. Under this construction, we enrich the varieties of valid quantum synchronizable codes. We also prove that the obtained quantum synchronizable codes can achieve maximum synchronization error tolerance. Furthermore, quantum synchronizable codes based on $(\lambda (u + v)|u - v)$ construction are shown to be able to have a better capability in correcting bit errors than those from projective geometry codes.

On the Hamming Distance of Repeated-Root Cyclic Codes of Length 6ps

Let $p$ be an odd prime, $s$ , $m$ be positive integers such that $p^{m}\equiv 2 \pmod 3$ . In this paper, using the relationship about Hamming distances between simple-root cyclic codes and repeated-root cyclic codes, the Hamming distance of all cyclic codes of length $6p^{s}$ over finite field $\mathbb F_{p^{m}}$ is obtained. All maximum distance separable (MDS) cyclic codes of length $6p^{s}$ are established.

Two new families of quantum synchronizable codes

In this paper, we present two new ways of quantum synchronization coding based on the $$(\varvec{u}+\varvec{v}|\varvec{u}-\varvec{v})$$ construction and the product construction, respectively, and greatly enrich the varieties of available quantum synchronizable codes. The circumstances where the maximum synchronization error tolerance can be reached are explained for both constructions. Furthermore, repeated-root cyclic codes derived from the $$(\varvec{u}+\varvec{v}|\varvec{u}-\varvec{v})$$ construction are shown to be able to provide better Pauli error-correcting capability than BCH codes.

Optimal b-symbol constacyclic codes with respect to the Singleton bound

Let [Formula: see text] be the finite field of order [Formula: see text], where [Formula: see text] is a power of odd prime [Formula: see text]. Assume that [Formula: see text], [Formula: see text] are nonzero elements of the finite field [Formula: see text] such that [Formula: see text]. In this paper, we determine the [Formula: see text]-distance of [Formula: see text]-constacyclic codes with generator polynomials [Formula: see text] of length [Formula: see text], where [Formula: see text] and [Formula: see text]. As an application, all maximum distance separable (MDS) [Formula: see text]-symbol constacyclic codes of length [Formula: see text] over [Formula: see text] are established. Among other results, we construct several classes of new MDS symbol-pair codes with minimum symbol-pair distance six or seven by using repeated-root cyclic codes of length [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is an odd prime.

Repeated-Root Constacyclic Codes of Length $$k\ell p^s$$

In this paper, we investigate all repeated-root constacyclic codes and their duals of length $$k\ell p^s$$ over $${\mathbb {F}}_q$$ in terms of generator polynomials, where $$\ell $$ is an odd prime different from p and k is an odd prime different from both $$\ell $$ and p such that $$k=4h+1$$ for some prime h. As an application, the characterization and enumeration of all self-dual repeated-root cyclic codes of length $$2^sk\ell $$ over $${\mathbb {F}}_q$$ are obtained.

Quantum Codes Constructed From Repeated-Root Cyclic Codes

In this paper, we use q-cyclotomic cosets modulo n = q − 1 and generator polynomial to describe repeated-root cyclic codes of length N = pαn. We determine the condition under which a repeated-root cyclic code contains its dual code. Then, for q = 2a, we give a class of dual-containing repeated-root cyclic codes. Applying CSS construction and Steane construction to these underlying classical codes, many quantum codes with good parameters shall be constructed, which are new in the sense that they are not available in the literature.

Non-Binary Quantum Synchronizable Codes From Repeated-Root Cyclic Codes

In this paper, we construct a new family of quantum synchronizable codes from repeated-root cyclic codes of lengths $p^{s}$ and $lp^{s}$ over $\mathbb {F}_{q}$ , where $s\geq 1~and l\geq 2$ are integers, and $p\geq 3$ is the odd characteristic. Within some loose limitations, these synchronizable codes can possess the best possible capability in synchronization recovery, and therefore, enriches the variety of good quantum synchronizable codes. Furthermore, by using known techniques in classical coding theory which convert the computation of the minimum distance of a repeated-root cyclic code to that of a shorter simple-root cyclic code, we prove that the repeated-root cyclic codes of lengths $p^{s}$ and $lp^{s}$ are in general better than narrow-sense BCH codes of close lengths in terms of minimum distances, and thereby enable the obtained synchronizable codes to correct more Pauli errors.

Uniform constant composition codes derived from repeated‐root cyclic codes

Uniform constant composition (UCC) codes are introduced, which are p-ary constant composition codes with block length mp where each p-ary symbol appears exactly m times in each codeword, being m a positive integer and p a prime. UCC codes are derived from a new class of p-ary repeated-root cyclic codes and two constructions are presented. Both constructions are at least asymptotically optimal with increasing p when compared with known best bounds for constant composition codes.

Constacyclic Symbol-Pair Codes: Lower Bounds and Optimal Constructions

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. MDS symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai {\it et al.} (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes.

On LCD repeated-root cyclic codes over finite fields

In this paper, we investigate the LCD repeated-root cyclic codes of length \(n=n'p^r\) over the finite field \(\mathbb {F}_q\), where gcd\((n',p)=1\). We give a necessary and sufficient condition for a repeated-root cyclic code to be LCD over \(\mathbb {F}_q\). We also determine the minimum distance of LCD repeated-root cyclic codes over \(\mathbb {F}_q\). Finally, we give the enumeration of LCD repeated-root cyclic codes of length n over \(\mathbb {F}_q\).