Projective two-weight irreducible cyclic and constacyclic codes

After several remarks on two-weight irreducible cyclic codes, we introduce a family of projective two-weight cyclic codes and a family of projective two-weight constacyclic codes and we discuss the existence of such codes.     

New classes of optimal ternary cyclic codes with minimum distance four

Cyclic code is an interesting topic in coding theory and communication systems. In this paper, we investigate the ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$ based on some results proposed by Ding and Helleseth in 2013. Six new classes of optimal ternary cyclic codes are presented by determining the solutions of certain equations over ${\mathbb{F}}_{3^m}$.     

A family of 2-weight codes related to BCH-codes

For every prime-power q and every pair of natural numbers m ≤ n′, we construct a q-ary linear code of length q^m (q^n′ − 1)(q^n′ − q^n′−m + 1)/(q − 1) and dimension 3n′, whose only nonzero weights are and . These code parameters and those of the corresponding family of strongly regular graphs are new in odd characteristic. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 391–396, 1997     

Primitive idempotents of irreducible cyclic codes and self-dual cyclic codes over Galois rings

Let $R$ be the Galois ring $GR(p^k,s)$ of characteristic $p^k$ and cardinality $p^{sk}$. Firstly, we give all primitive idempotent generators of irreducible cyclic codes of length $n$ over $R$, and a $p$-adic integer ring with $gcd(p,n)=1$. Secondly, we obtain all primitive idempotents of all irreducible cyclic codes of length $r{l}^m$ over $R$, where $r,l,$ and $t$ are three primes with $2?l$, $r|(q^t?1)$, $l^v\parallel (q^t?1)$ and $gcd(rl,q(q?1))=1$. Finally, as applications, weight distributions of all irreducible cyclic codes for $t=2$ and generator polynomials of self-dual cyclic codes of length $l^m$ and $r{l}^m$ over $R$ are given.     

The weight distributions of irreducible cyclic codes of length

Let m be a positive integer and q be an odd prime power. In this paper, the weight distributions of all the irreducible cyclic codes of length 2^m over F_q are determined explicitly.     

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.     

Several classes of optimal ternary cyclic codes with minimal distance four

In this paper, by analyzing the solutions of certain equations over ${\mathbb{F}}_{3^m}$, we present four classes of optimal ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$. It is shown that some recent work on this class of optimal ternary cyclic codes are special cases of our results.     

The Correspondence Between Projective Codes and 2-weight Codes

We show how to get a 1-1 correspondence between projective linear codes and 2-weight linear codes. A generalization of the construction gives rise to several new ternary linear codes of dimension six.     

Roos bound for skew cyclic codes in Hamming and rank metric

In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct codes with a prescribed minimum rank distance, using the skew Roos bound, is also given. Moreover, some examples of MDS codes and MRD codes over finite fields are built, using the skew Roos bound.     

Cyclically permutable representations of cyclic codes

Cyclically permutable codes have been studied for several applications involving synchronization, code-division multiple-access (CDMA) radio systems and optical CDMA. The usual emphasis is on finding constant weight cyclically permutable codes with the maximum number of codewords. In this paper the question of when a particular error-correcting code is equivalent (by permutation of the symbols) to a cyclically permutable code is addressed. The problem is introduced for simplex codes and a motivating example is given. In the final section it is shown that the construction technique may be applied in general to cyclic codes.     

A class of minimal cyclic codes over finite fields

Let ${\mathbb{F}}_q$ be a finite field of odd order $q$ and $n=2^a{p}_1^{a_1}{p}_2^{a_2}$, where $a,a_1,a_2$ are positive integers, $p_1,p_2$ are distinct odd primes and $4p_1{p}_2|q?1$. In this paper, we study the irreducible factorization of $x^n?1$ over ${\mathbb{F}}_q$ and all primitive idempotents in the ring ${\mathbb{F}}_q\left[x\right]∕〈x^n?1〉$.Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length $n$ over ${\mathbb{F}}_q$.