A characterization of all semiprimitive irreducible cyclic codes in terms of their lengths

Let Fq be a finite field with q elements and n a positive integer. In this paper, we use matrix method to give all primitive idempotents of irreducible cyclic codes of length n, whose prime divisors divide q-1.

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Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

Mathematical Problems in Engineering ◽

10.1155/2016/9020173 ◽

2016 ◽

Vol 2016 ◽

pp. 1-12 ◽

Cited By ~ 4

Author(s):

Parinyawat Choosuwan ◽

Somphong Jitman ◽

Patanee Udomkavanich

Keyword(s):

Cyclic Codes ◽

Group Algebras ◽

Dual Codes ◽

Complete Enumeration ◽

Odd Order ◽

Abelian Codes ◽

Positive Integers ◽

Ideal Group ◽

Suitable Product

The main focus of this paper is the complete enumeration of self-dual abelian codes in nonprincipal ideal group algebrasF2k[A×Z2×Z2s]with respect to both the Euclidean and Hermitian inner products, wherekandsare positive integers andAis an abelian group of odd order. Based on the well-known characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length2sover some Galois extensions of the ringF2k+uF2k, whereu2=0. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of lengthpsoverFpk+uFpkare given. Combining these results, the complete enumeration of self-dual abelian codes inF2k[A×Z2×Z2s]is therefore obtained.

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Characterization of cyclic codes over {ℬ[X;(1/m)Z0]}m > 1and efficient encoding decoding algorithm for cyclic codes

International Journal of Computer Mathematics ◽

10.1080/00207160.2016.1158815 ◽

2016 ◽

Vol 94 (5) ◽

pp. 1015-1027

Author(s):

Tariq Shah ◽

Naveed Ahmed Azam

Keyword(s):

Cyclic Codes ◽

Decoding Algorithm

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Two-Weight Codes Punctured from Irreducible Cyclic Codes

Coding and Cryptology ◽

10.1142/9789812832245_0009 ◽

2008 ◽

Cited By ~ 25

Author(s):

Cunsheng Ding ◽

Jinquan Luo ◽

Harald Niederreiter

Keyword(s):

Cyclic Codes ◽

Irreducible Cyclic Codes

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Skew cyclic codes over 𝔽4R

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500657 ◽

2020 ◽

pp. 2250065

Author(s):

Nasreddine Benbelkacem ◽

Martianus Frederic Ezerman ◽

Taher Abualrub ◽

Nuh Aydin ◽

Aicha Batoul

Keyword(s):

Error Control ◽

Cyclic Codes ◽

Gray Map ◽

Inner Product ◽

Algebraic Structures ◽

Error Control Codes ◽

Dna Codes ◽

Natural Connection ◽

Skew Cyclic Codes

This paper considers a new alphabet set, which is a ring that we call [Formula: see text], to construct linear error-control codes. Skew cyclic codes over this ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize [Formula: see text]-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over [Formula: see text] are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of [Formula: see text]-skew cyclic codes which are reversible complement.

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