Prime Power Length Research Articles

Symbol-Triple Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over Fq+uFq+u2Fq

Let p be an odd prime, where ϑ and m are positive integers. Let ψ be a nonzero element of the finite field Fq, where q=pm, and R=Fq+uFq+u2Fq(u3=0). In this paper, we determine completely the symbol-triple distances of all ψ-constacyclic codes of length pϑ over R.

Duality of constacyclic codes of prime power length over the finite non-commutative chain ring [formula omitted

Let R=Fpm[u,θ]〈u2〉 where θ is an automorphism of Fpm but it is not the identity. The list of (left and right) α-constacyclic codes of length ps over R is provided where θ(α)=α. The self-dual α-constacyclic codes are given. In addition, we derive the condition of LCD codes for those codes. For the remaining result, the condition of self-orthogonal left α-constacyclic codes is also obtained.

Roulette games and depths of words over finite commutative rings

In this paper, we propose three new turn-based two player roulette games and provide positional winning strategies for these games in terms of depths of words over finite commutative rings with unity. We further discuss the feasibility of these winning strategies by studying depths of codewords of all repeated-root $$(\alpha +\gamma \beta )$$ -constacyclic codes of prime power lengths over a finite commutative chain ring $${\mathcal {R}},$$ where $$\alpha $$ is a non-zero element of the Teichmuller set of $${\mathcal {R}},$$ $$\gamma $$ is a generator of the maximal ideal of $${\mathcal {R}}$$ and $$\beta $$ is a unit in $${\mathcal {R}}.$$ As a consequence, we explicitly determine depth distributions of all repeated-root $$(\alpha +\gamma \beta )$$ -constacyclic codes of prime power lengths over $${\mathcal {R}}$$ .

MDS Constacyclic Codes of Prime Power Lengths Over Finite Fields and Construction of Quantum MDS Codes

If we fix the code length n and dimension k, maximum distance separable (briefly, MDS) codes form an important class of codes because the class of MDS codes has the greatest error-correcting and detecting capabilities. In this paper, we establish all MDS constacyclic codes of length ps over $\mathbb {F}_{p^{m}}$ . We also give some examples of MDS constacyclic codes over finite fields. As an application, we construct all quantum MDS codes from repeated-root codes of prime power lengths over finite fields using the CSS and Hermitian constructions. We provide all quantum MDS codes constructed from dual codes of repeated-root codes of prime power lengths over finite fields using the Hermitian construction. They are new in the sense that their parameters are different from all the previous constructions. Moreover, some of them have larger Hamming distances than the well known quantum error-correcting codes in the literature.

B-Symbol Distance of Constacylic Codes of Length ps Over Fp m + uFp m

In this research paper, the repeated-root constacyclic codes over the chain ring F p m + uF p m are considered, where p is a prime and m > 0 is any integer. The b-symbol distance for prime power length, i.e. p s is also studied for any integer s > 0. The Hamming and symbol-pair distances of all δ-constacyclic codes have been thoroughly studied in [18], where δ is an unit in the ring F p m + uF p m which is of the form ζ and φ + uφ, where 0 ≠ 6 φ, φ, ζ ∈ F p m. In this paper, the b-symbol distance of all such δ-constacyclic codes of prime power length is computed for 1 ≤ b ≤ ⌊p/2⌋. Furthermore, as an application, all MDS b-symbol constacyclic codes of length p s over F p m + uF p m are established.

Symbol-triple distance of repeated-root constacyclic codes of prime power lengths

Let [Formula: see text] be an odd prime, [Formula: see text] and [Formula: see text] be positive integers and [Formula: see text] be a nonzero element of [Formula: see text]. The [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] are linearly ordered under set theoretic inclusion as ideals of the chain ring [Formula: see text]. Using this structure, the symbol-triple distances of all such [Formula: see text]-constacyclic codes are established in this paper. All maximum distance separable symbol-triple constacyclic codes of length [Formula: see text] are also determined as an application.

Generalized cyclotomic numbers and cyclic codes of prime power length over Z4

Generalized cyclotomic numbers of order [Formula: see text] with respect to an odd prime power are obtained. Hence, explicit expressions for primitive idempotents in the ring [Formula: see text] are obtained in two cases, when the multiplicative order of 2 modulo [Formula: see text] is [Formula: see text] and [Formula: see text], where [Formula: see text] is an odd prime. Orthogonality and self-duality of some [Formula: see text] cyclic codes are also discussed. Further, a method for obtaining cyclic self-dual/isodual codes of length [Formula: see text] over [Formula: see text] is given.

On the Structure and Distances of Repeated-Root Constacyclic Codes of Prime Power Lengths Over Finite Commutative Chain Rings

Let p be a prime, s be a positive integer, and R be a finite commutative chain ring with the characteristic as a power of p. For a unit λ ε R, λ-constacyclic codes of length ps over R are ideals of the quotient ring R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> -λ). In this paper, we derive necessary and sufficient conditions under which the quotient ring R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a chain ring. When R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a chain ring, all λ-constacyclic codes of length ps over R are known. In this paper, we establish the algebraic structures of all λ-constacyclic codes of length ps over R when R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a non-chain ring. We also determine the number of codewords in each of these codes. Using their algebraic structures, we obtain symbol-pair distances, Rosenbloom-Tsfasman (RT) distances, and RT weight distributions of all constacyclic codes of length ps over R. Apart from this, we derive necessary and sufficient conditions under which a constacyclic code of length ps over R is maximumdistance separable with respect to the: 1) Hamming metric; 2) symbol-pair metric; and 3) RT metric. We also provide an algorithm to decode the constacyclic codes of length ps over R using the known decoding algorithms of linear codes over finite fields with respect to the Hamming, symbol-pair, and RT metrics.

Cyclic codes of prime power length from generalized cyclotomic classes of order 2r

In this paper, we first introduce generalized cyclotomic classes of order 2r, r ≥ 1 and then present a special class of cyclic codes with length pm. We also obtain lower bound on the minimum odd weight of these codes.

On the Symbol-Pair Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths

Let $p$ be a prime, and $\lambda$ be a nonzero element of the finite field $\mathbb F_{p^{m}}$ . The $\lambda$ -constacyclic codes of length $p^{s}$ over $\mathbb F_{p^{m}}$ are linearly ordered under set-theoretic inclusion, i.e., they are the ideals $\langle (x-\lambda _{0})^{i} \rangle$ , $0 \leq i \leq p^{s}$ of the chain ring $[({\mathbb F_{p^{m}}[x]})/({\langle x^{p^{s}}-\lambda \rangle })]$ . This structure is used to establish the symbol-pair distances of all such $\lambda$ -constacyclic codes. Among others, all maximum distance separable symbol-pair constacyclic codes of length $p^{s}$ are obtained.

English

English

The weight distribution for any irreducible cyclic code of length $$p^m$$ p m

We generalize and simplify the results of Sharma and Bakshi (Finite Fields Appl 18(1):144–159 2012) on the weight-distributions of irreducible cyclic codes of prime-power length.

On the equivalence of cyclic and quasi-cyclic codes over finite fields

This paper studies the equivalence problem for cyclic codes of length $p^r$ and quasi-cyclic codes of length $p^rl$. In particular, we generalize the results of Huffman, Job, and Pless (J. Combin. Theory. A, 62, 183--215, 1993), who considered the special case $p^2$. This is achieved by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. This allows us to obtain an algorithm which solves the problem of equivalency for cyclic codes of length $p^r$ in polynomial time. Further, we characterize the set by which two quasi-cyclic codes of length $p^rl$ can be equivalent, and prove that the affine group is one of its subsets.

Repeated-root constacyclic codes of prime power lengths over finite chain rings

We study the algebraic structure of repeated-root λ-constacyclic codes of prime power length ps over a finite commutative chain ring R with maximal ideal 〈γ〉. It is shown that, for any unit λ of the chain ring R, there always exists an element r∈R such that λ−rps is not invertible, and furthermore, the ambient ring R[x]〈xps−λ〉 is a local ring with maximal ideal 〈x−r,γ〉. When there is a unit λ0 such that λ=λ0ps, the nilpotency index of x−λ0 in the ambient ring R[x]〈xps−λ〉 is established. When λ=λ0ps+γw, for some unit w of R, it is shown that the ambient ring R[x]〈xps−λ〉 is a chain ring with maximal ideal 〈xps−λ0〉, which in turn provides structure and sizes of all λ-constacyclic codes and their duals. Among other things, situations when a linear code over R is both α- and β-constacyclic, for different units α, β, are discussed.

Repeated-root constacyclic codes of prime power length over [formula omitted] and their duals

The units of the chain ring Ra=Fpm[u]〈ua〉=Fpm+uFpm+⋯+ua−1Fpm are partitioned into a distinct types. It is shown that for any unit Λ of Type k, a unit λ of Type k∗ can be constructed, such that the class of λ-constacyclic of length ps of Type k∗ codes is one-to-one correspondent to the class of Λ-constacyclic codes of the same length of Type k via a ring isomorphism. The units of Ra of the form Λ=Λ0+uΛ1+⋯+ua−1Λa−1, where Λ0,Λ1,…,Λa−1∈Fpm, Λ0≠0,Λ1≠0, are considered in detail. The structure, duals, Hamming and homogeneous distances of Λ-constacyclic codes of length ps over Ra are established. It is shown that self-dual Λ-constacyclic codes of length ps over Ra exist if and only if a is even, and in such case, it is unique. Among other results, we discuss some conditions when a code is both α- and β-constacyclic over Ra for different units α, β.

Some cyclic codes of prime-power length

A new class of cyclic codes of length 2n over GF(q) is proposed, where q is a prime of the form 8m ± 3 and n > 3 is an integer. These codes are defined in terms of their generator polynomials. These codes have many properties analogous to those of duadic codes. Generator polynomials of some duadic codes of length pn over GF(q) are also discussed, where p is an odd prime, n is an integer and q = ρ or ρ2 for some prime ρ.

Brauer Characters of Prime Power Degrees and Conjugacy Classes of Prime Power Lengths

We study the structure of any finite group in which the degree of every irreducible p-Brauer character, respectively the length of every p-regular conjugacy class, is a power of a fixed prime.

Polyadic codes of prime power length

Polyadic codes constitute a special class of cyclic codes and are generalizations of quadratic residue codes, duadic codes, triadic codes, m-adic residue codes and split group codes, which have good error-correcting properties. In this paper, we give necessary and sufficient conditions for the existence of polyadic codes of prime power length. Examples of some good codes arising from the family of polyadic codes of prime power length are also given.

Idempotents in group algebras and minimal abelian codes

We compute the number of simple components of a semisimple finite abelian group algebra and determine all cases where this number is minimal; i.e. equal to the number of simple components of the rational group algebra of the same group. This result is used to compute idempotent generators of minimal abelian codes, extending results of Arora and Pruthi [S.K. Arora, M. Pruthi, Minimal cyclic codes of length 2 p n , Finite Field Appl. 5 (1999) 177–187; M. Pruthi, S.K. Arora, Minimal codes of prime power length, Finite Field Appl. 3 (1997) 99–113]. We also show how to compute the dimension and weight of these codes in a simple way.

A simple proof of the multiplicativity of directed cycles of prime power length

This paper gives a simple combinatorial proof of the multiplicativity of directed cycles of prime power length.