Negacyclic Codes Of Length Research Articles

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

Negacyclic codes of length [Formula: see text] over the Galois ring [Formula: see text] are linearly ordered under set-theoretic inclusion, i.e., they are the ideals [Formula: see text], [Formula: see text], of the chain ring [Formula: see text]. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field [Formula: see text] (i.e., [Formula: see text]), the symbol-pair distance distribution of constacyclic codes over [Formula: see text] verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length [Formula: see text] over [Formula: see text].

Explicit constructions of cyclic and negacyclic codes of length 3ps over 𝔽pm + u𝔽pm

Let [Formula: see text] be a prime such that [Formula: see text]. The algebraic structures of all cyclic and negacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text] are obtained that the conditions depend on the factorization of polynomial [Formula: see text] over [Formula: see text]. Therefore, we classify the structures of cyclic and negacyclic codes of length [Formula: see text] over [Formula: see text] into 2 cases, i.e., [Formula: see text] and [Formula: see text]. From that we obtain the number of all cyclic and negacyclic codes of length [Formula: see text] over [Formula: see text]. After that, we give some situations for such cyclic and negacyclic codes are self-dual codes.

A note on self-dual negacyclic codes of length $$p^s$$ over "Equation missing"

Self-dual cyclic codes over rings and their generalizations have become of interest due to their rich algebraic structures and wide applications. Cyclic and self-dual cyclic codes over the ring have been quite well studied, where p is a prime, k is a positive integer, and $$u^2=0$$ . We focus on negacyclic codes over , where p is an odd prime and k is a positive integer. An alternative and explicit algebraic characterization of negacyclic codes of length $$p^s$$ over is presented. Based on this result, representation and enumeration of self-dual negacyclic codes of length $$p^s$$ over are given under both the Euclidean and Hermitian inner products.

On cyclic and negacyclic codes of length 8ℓmpn over finite field

Let [Formula: see text] be finite field with [Formula: see text] elements, where [Formula: see text] be power of an odd prime [Formula: see text] In this paper, we determine generator polynomials of all cyclic and negacyclic codes of length [Formula: see text] over [Formula: see text] where [Formula: see text] are distinct odd primes and [Formula: see text] are positive integers. We also determine all self-dual, self-orthogonal, complementary-dual cyclic and negacyclic codes of length [Formula: see text] over [Formula: see text]

Cyclic and negacyclic codes of length 4ps over 𝔽pm + u𝔽pm

Let [Formula: see text] be the finite field of order [Formula: see text] where [Formula: see text] is an odd prime and [Formula: see text] is a positive integer. In this paper, we determine the algebraic structures of all cyclic and negacyclic codes of length [Formula: see text] over the finite commutative chain ring [Formula: see text] where [Formula: see text] and [Formula: see text] is a positive integer. We also obtain the number of codewords in each of these codes. Among others, we establish the duals of all such codes and derive some self-dual cyclic and negacyclic codes of length [Formula: see text] over [Formula: see text]

The Average Hull Dimension of Negacyclic Codes over Finite Fields

Hulls of linear codes have been extensively studied due to their wide applications and links with the efficiency of some algorithms in coding theory. In this paper, the average dimension of the Euclidean hull of negacyclic codes of length n over finite fields F q , denoted by E ( n , − 1 , q ) , has been investigated. The formula for E ( n , − 1 , q ) has been determined. Some upper and lower bounds of E ( n , − 1 , q ) have been given as well. Asymptotically, it has been shown that either E ( n , − 1 , q ) is zero or it grows the same rate as n.

Repeated-root constacyclic codes of arbitrary lengths over the Galois ring GR(p2,m)

Let [Formula: see text] be a prime, [Formula: see text] be a positive integer, and let GR[Formula: see text] be the Galois ring of characteristic [Formula: see text] and cardinality [Formula: see text]. In this paper, all repeated-root constacyclic codes of arbitrary lengths over GR[Formula: see text] their sizes and their dual codes are determined. As an application, some isodual constacyclic codes over GR[Formula: see text] are also listed. To illustrate the results, all cyclic and negacyclic codes of length 10 over [Formula: see text] are obtained.

Negacyclic codes of length 2k over

ABSTRACTIn this paper, we study negacyclic codes of length over the ring , . We give the complete structure of negacyclic codes and determine their cardinalities. The structure of the duals of negacyclic codes is determined and self-dual negacyclic codes are characterized. We present some new good liner codes which are the Gray images of negacylic codes over R.

Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields

The main objective of this article is to study self-orthogonal negacyclic codes of length $n$ over a finite field $\mathbb{F}_{q}$, where the characteristic of $\mathbb{F}_{q}$ does not divide $n$. We investigate issues related to their existence, characterization and enumeration. We find the necessary and sufficient conditions for the existence of self-orthogonal negacyclic codes of length $n$ over a finite field $\mathbb{F}_{q}$. We characterize the defining sets and the corresponding generator polynomials of these codes. We obtain formulae to calculate the number of self-dual and self-orthogonal negacyclic codes of a given length $n$ over $\mathbb{F}_{q}$. The enumeration formula for self-orthogonal negacyclic codes involves a two-variable function $\chi(d,q)$ defined by $\chi(d,q)=0$ if $d$ divides $(q^{k}+1)$ for some $k\geq0$ and $\chi(d,q)=1$, otherwise. We give necessary and sufficient conditions when $\chi(d,q)=0$ holds.

Hulls of cyclic and negacyclic codes over finite fields

We study the hulls of cyclic and negacyclic codes of length n over a finite field Fq with respect to the Euclidean and Hermitian inner products. Based on the characterization of their generator polynomials, the dimensions of the hulls of cyclic and negacyclic codes of length n over Fq are determined. The enumerations of cyclic codes and negacyclic codes of length n over Fq having hulls of a given dimension are established as well. As special cases, known results concerning complementary dual and self-dual cyclic codes can be viewed as corollaries of results in this paper.

Construction of repeated-root constacyclic code of length 8 p s over $F_{p^m } $

For any odd prime p, a classification of all cyclic and negacyclic codes of length 8 ps over \(F_{p^m } \) are obtained, which establishes the algebra structures in term of specified polynomial generators of such codes. Among other results, all self-dual negacyclic codes of length 8 ps are obtained, and the structures of α-constacyclic and β-constacyclic codes of length 8 ps over \(F_{p^m } \) are established.

Repeated-root constacyclic codes of length ℓ t p s and their dual codes

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 $\mathbb {F}_{7}$ .

The Hamming Distances of Negacyclic Codes of Length 2 s over GR(2 a ,m)

The Hamming distances of all negacyclic codes of length 2 s over the Galois ring GR(2 a ,m) are given. In particular, the Lee distances of all negacyclic codes over Z 4 of length 2 s are obtained. The Gray images of such negacyclic codes over Z 4 are also determined under the Gray map.

On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions

We investigate negacyclic and cyclic codes of length p s over the finite field F p a . Negacyclic codes of length p s are precisely the ideals of the chain ring F p a [ x ] 〈 x p s + 1 〉 . This structure is then used to obtain the Hamming distance distribution of the class of such negacyclic codes, which also provides Hamming weight distributions and enumerations of several codes. An one-to-one correspondence between negacyclic and cyclic codes is established to carry accordingly those results of negacyclic codes to cyclic codes.