Linear Cyclic Codes Research Articles

Cyclic LRC codes, binary LRC codes, and upper bounds on the distance of cyclic codes

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalize the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, no. 8, 2014). In this paper we focus on optimal cyclic codes that arise from this construction. We give a characterization of these codes in terms of their zeros, and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance. The locality parameter of a cyclic code is related to the dual distance of this code, and we phrase our results in terms of upper bounds on the dual distance.

Efficient Decoding of Short Length Linear Cyclic Codes

Iterative soft decision decoding of linear block codes is a practical necessity when working with even modest block lengths. A number of algorithms have been proposed in the literature which use the permutation group of a code and the belief propagation (BP) algorithm for decoding. A novel soft-input, soft-output algorithm is presented that can be used for efficiently decoding of linear cyclic codes. Utilising the automorphism property of cyclic codes the permutation is incorporated into the belief propagation algorithm resulting in faster convergence and better error correcting performance. Performance of the new approach is analysed using a (63,45) BCH code and a (72,36) quadratic residue code. Simulation results show significant reduction in the average number of required decoding iterations and some improvement in error correcting performance over published algorithms.

The Gray images of $$(1+u)$$ ( 1 + u ) constacyclic codes over $$F_{2^m}[u]/\langle u^{k} \rangle $$ F 2 m [ u ] / ⟨ u k ⟩

Let $$R_{k}$$ denote the polynomial residue ring $$F_{2^m}[u]/\langle u^{k} \rangle $$ , where $$2^{j-1}+1\le k\le 2^{j}$$ for some positive integer $$j$$ . Motivated by the work in [1], we introduce a new Gray map from $$R_{k}$$ to $$F_{2^m}^{2^{j}}$$ . It is proved that the Gray image of a linear $$(1+u)$$ constacyclic code of an arbitrary length $$N$$ over $$R_{k}$$ is a distance invariant linear cyclic code of length $$2^{j}N$$ over $$F_{2^m}$$ . Moreover, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over $$F_{2}$$ and $$F_{4}$$ are constructed under this Gray map.

The Gray image of a class of constacyclic codes over polynomial residue rings

Let Rk denote the polynomial residue ring F2m[u]/〈uk〉, where 2j−1+1≤k≤2j for some positive integer j. Motivated by the work in Abular and Siap ([1], 2009) and Qian ([8], 2011), we introduce a new Gray map from Rk to F2m2j. It is proved that the Gray image of a linear (1+u+⋯+uk−1) constacyclic code of an arbitrary length N over Rk is a distance invariant linear cyclic code of length 2jN over F2m. Furthermore, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over F2 and F4 are constructed under this Gray map.

Generalized residue and t-residue codes and their idempotent generators

In this paper a general class of linear cyclic codes $$C_{n,q,t}^i , 1\le i\le t$$ C n , q , t i , 1 ≤ i ≤ t , is defined of length $$n$$ n and over a field $${ GF}(q)$$ G F ( q ) with $$(n,q)=1$$ ( n , q ) = 1 . This class of codes includes as special cases quadratic residue codes, generalized quadratic residue codes, $$e$$ e -residue codes and $$Q$$ Q -codes. Furthermore, they partially overlap with the families of duadic, triadic and polyadic codes. Expressions for idempotent generators are derived in terms of the size of cyclotomic cosets mod $$n $$ n and coefficients of the irreducible polynomials over $${ GF}(q)$$ G F ( q ) dividing $$x^{n}-1$$ x n - 1 . As an auxiliary tool an orthonormal matrix is introduced whose columns correspond to these idempotents. Concrete examples are presented for $$t=2$$ t = 2 and $$n\in \{p^{\lambda },2p^{\lambda },2^{\lambda }\}, \lambda \ge 1$$ n ? { p ? , 2 p ? , 2 ? } , ? ? 1 , where $$p$$ p is an arbitrary odd prime. When $$n=p^{\lambda }$$ n = p ? or $$n=2p^{\lambda }$$ n = 2 p ? the codes all belong to the subclass of 2-residue codes. Using this technique, we determine the idempotents of the codes $$C_{2^{\lambda },q,2}^i $$ C 2 ? , q , 2 i , and recover those of the generalized quadratic codes $$C_{p^{\lambda },q,2}^i $$ C p ? , q , 2 i and of the codes $$C_{2p^{\lambda },q,2}^i $$ C 2 p ? , q , 2 i . In the final section the idempotents of the cubic residue codes $$C_{p,q,3}^i $$ C p , q , 3 i are constructed.

Unified combinatorial constructions of optimal optical orthogonal codes

We present unified constructions of optical orthogonalcodes (OOCs) using other combinatorial objects such as cyclic linear codes and frequency hopping sequences.Some of the obtained OOCs are optimal or asymptotically optimal with respect tothe Johnson bound. Also, we are able to show the existence of new optimal frequency hopping sequences (FHSs) with respect to the Singleton bound from our observation on a relation between OOCs and FHSs.The last construction is based on residue rings of polynomials over finite fields, and it yields a new large class of asymptotically optimal $(q-1,k,k-2)$-OOCs for any prime power $q$ with $\gcd{(q-1,k)}=1$. Some infinite families of optimal ones are included as a subclass.

An Efficient Encoding Algorithm for (n, k) Binary Cyclic Codes

In forward error correction (FEC) systems, basically two mechanisms namely block coding and convolutional coding are used for error-control coding. The error-control coding processes involve encoding and decoding information symbols to get rid of errors produced by noise in communication channels. In a binary block encoding technique, message words are arranged in blocks of k bits which are converted into code words of n bits, n> k , by adding redundancy bits. Encoding is a compulsory part of error correcting codes and without knowledge of encoding structure the decoding of code words is impossible. In this paper a very simple and most effective technique for systematic encoding of linear binary cyclic codes is used which improves the encoding speed by reducing the computational complexity. In addition to encoding speed, the encoding can be done by hand using XOR (modulo-2) operations without involving computers after generating an encoding table for the code.

A Class of Constacyclic Codes over <i>R</i> + <i>vR</i> and Its Gray Image

We study (1 + 2v)-constacyclic codes overR + vR and their Gray images, where v2 + v = 0 and R is a finite chain ring with maximal ideal λ> and nilpotency index e. It is proved that the Gray map images of a (1 + 2v)-constacyclic codes of length n over R + vR are distance-invariant linear cyclic codes of length 2n over R. The generator polynomials of this kind of codes for length n are determined, where n is relatively prime to p, p is the character of the field R/λ> . Their dual codes are also discussed.

미지의 선형 순회부호에 대한 복원기법

잡음에 있는 채널을 통한 디지털 통신에서는 채널 잡음에 대항하기 위해 오류정정부호(채널부호)를 사용하게 된다. 만일 송신측의 협조 없이 전송정보를 알아내려면 사용된 채널부호를 복원하는 것이 무엇보다 중요하다. 본 논문에서는 잡음에 오염된 수신 비트열로부터 사용된 채널부호의 여려 파라메타를 추출하여 궁극적으로 채널부호를 복원하는 채널부호 복원기법 중 순회부호(cyclic code)의 복원 기법을 제안한다. In most digital communication systems over the noisy channel, some form of forward error correction scheme is employed for reliable communications. If one wants to recover the transmitted message without any knowledge of the error correcting codes employed, it is of utmost importance to figure out and reconstruct the error correcting codes. In this paper, we propose two algorithms of reconstructing linear cyclic codes from the corrupted received bit sequence, one for general linear binary cyclic codes and the other for Reed-Solomon codes. For two algorithms, we ran computer simulations and the performances are shown to be superior to those with the conventional LWM method.

A class of constacyclic codes over [formula omitted] and its Gray image

We introduce a Gray map from F p + v F p to F p 2 and study ( 1 − 2 v ) -constacyclic codes over F p + v F p , where v 2 = v . It is proved that the image of a ( 1 − 2 v ) -constacyclic code of length n over F p + v F p under the Gray map is a distance-invariant linear cyclic code of length 2 n over F p . The generators of such constacyclic codes for an arbitrary length are determined and their dual codes are also discussed.

On some special codes over Fp+uFp+u2Fp

Abstract We define a new map between codes over F p + uF p + u 2 F p and F p which is different to that defined in [2] . It is proved that the image of the linear cyclic code over the commutative ring F p + uF p + u 2 F p with length n under this map is a distance-invariant quasi-cyclic code of index p 2 with length p 2 n over F p . Moreover, it is proved that, if ( n , p ) = 1, then every code with length p 2 n over F p which is the image of a linear (1 − u 2 )-cyclic code with length n over F p + uF p + u 2 F p under this map is permutation equivalent to a quasi-cyclic code of index p 2 .

Optimal Sets of Frequency Hopping Sequences From Linear Cyclic Codes

In communication systems, frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are used in FH-CDMA systems. In this paper, an earlier idea of constructing optimal sets of frequency hopping sequences is further investigated. New optimal parameters of sets of frequency hopping sequences are obtained with subcodes of the Reed-Solomon codes. Optimal sets of frequency hopping sequences are constructed with a class of irreducible cyclic codes. As a byproduct, the weight distribution of a subclass of irreducible cyclic codes is determined.

Design of parallel fault-secure encoders for systematic cyclic block transmission codes

In this paper, we consider the problem of designing parallel fault-secure encoders for various systematic cyclic linear codes used in data transmission. It is assumed that the data to be encoded before transmission are stored in a fault-tolerant RAM memory system protected against errors using a cyclic linear error detecting and/or correcting code. The main idea relies on taking advantage of the RAM check bits to control the correct operation of the cyclic code encoder as well. A slightly modified encoder allows not only for encoding the transmission data stream but also, independently and in parallel, to generate the reference check bits which allow for concurrent error detection in the encoder itself. The error detection capacity proves to be effective and grants good levels of protection as shown by error injection campaigns on encoders for various standard linear cyclic error detecting and error correcting codes. Moreover, the complexity evaluation of the FPGA implementations of the encoders shows that their fault-secure versions compare favorably against the unprotected ones, both with respect to hardware complexity and the maximal frequency of operation.

The gray image of the cyclic codes over

It is proved that every binary code which is the Gray image of a linear cyclic code over with length n is equivalent to a linear cyclic code, if n is odd.

On (1 − u 2)-cyclic and cyclic codes over IFp + uIFp + u 2 IFp

It is extended a result of [5] to codes over the commutative ring IFp + uIFp + u 2 IFp where u 3 = 0. A new Gray map between codes over IFp + uIFp + u 2 IFp and IFp is defined. It is proved that the Gray image of the linear (1 − u 2)-cyclic code over the commutative ring IFp + uIFp + u 2 IFp of length n is a distance-invariant quasi-cyclic code of index p and length p 2 n over IFp . And it is proved that if (n, p) = 1, then every code of length p 2 n over IFp which is the Gray image of a linear cyclic code of length n over IFp + uIFp + u 2 IFp is permutation equivalent to a quasi-cyclic code of index p.

A note on cyclic codes over [formula omitted] of length [formula omitted

Linear cyclic codes of length p k over the Galois ring GR ( p 2 , m ) , that is ideals of the ring GR ( p 2 , m ) [ u ] / 〈 u p k − 1 〉 , are studied. The form of the dual codes is analyzed and self-dual codes are identified.

Constacyclic codes as invariant subspaces

Constacyclic codes are generalizations of the familiar linear cyclic codes. In this paper constacyclic codes over a finite field F are regarded as invariant subspaces of F n with respect to a suitable linear operator. By applying standard techniques from linear algebra one can derive properties of these codes which generalize several well-known results for cyclic codes, such as the various lower bounds for the minimum distance.

Analysis of the interaction between linear cyclic error correcting codes and self-synchronous payload scramblers

In order to protect public network data transmission from potential Layer 1 attacks by malicious users, self-synchronous scramblers have come into widespread use. Such networks include those using ATM, packet over SONET (POS), and the new generic framing procedure (GFP). Unfortunately, feedback taps inherent in self-synchronous descramblers cause multiplication of transmission errors, which in turn degrades the performance of most linear cyclic error detection/correction codes. This paper analyzes this scrambler/code interaction with respect to the resulting probability of undetectable errors and transmission error correction capability. The theoretical criteria are derived for a linear cyclic code to maintain its error detection and correction performance in the presence of the scramblers. A novel approach for improving the error correction capabilities is also presented.

On (1− u) -cyclic codes over [formula omitted

We extend the results of [J.F. Qian, L.N. Zhang, S.X. Zhu, ( 1 + u ) -constacyclic and cyclic codes over F 2 + u F 2 , Appl. Math. Lett. 19 (2006) 820–823. [3]] to codes over the commutative ring R = F p k + u F p k , where p is prime, k ∈ N and u 2 = 0 . In particular, we prove that the Gray image of a linear ( 1 − u ) -cyclic code over R of length n is a distance-invariant quasicyclic code of index p k − 1 and length p k n over F p k . We also prove that if ( n , p ) = 1 , then every code of length p k n over F p k which is the Gray image of a linear cyclic code of length n over R is permutation-equivalent to a quasicyclic code of index p k − 1 .

Constacyclic and Cyclic Codes over F2 + uF2 + u2F2

A new Gray map between codes over F2 + uF2 + u2F2 and codes over F2 is defined. We prove that the Gray image of a linear (1-u2)-cyclic code over F2 + uF2 + u2F2 of length n is a binary distance invariant linear quasi-cyclic code. We also prove that, if n is odd, then every binary code which is the Gray image of a linear cyclic code over F2 + uF2 + u2F2 of length n is equivalent to a quasi-cyclic code.