#### Title

Repeated-Root Constacyclic Codes of Prime Power Length Over Fpm [u] / 〈ua〉 and Their Duals

#### Publication Title

Discrete Mathematics

#### Publication Date

6-2016

#### Document Type

Article

#### DOI

10.1016/J.DISC.2016.01.020

#### Keywords

constacyclic codes, dual codes, chain rings, polynomial residue rings, hamming distance, homogeneous distance

#### Disciplines

Algebra | Discrete Mathematics and Combinatorics | Mathematics

#### Abstract

The units of the chain ring ℛ_{a} = **F**_{pm} [*u*]/〈*u ^{a}*〉 =

**F**

_{pm}+

*u*

**F**

_{pm}+ ⋯ +

*u*

^{a}^{−1}

**F**

_{pm}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

*p*of Type

^{s}*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 ℛ

_{a}of the form

*Λ*=

*Λ*

_{0}+

*u*

*Λ*

_{1}+ ⋯ +

*u*

^{a}^{−1}

*Λ*

_{a}_{−1}, where

*Λ*

_{0},

*Λ*

_{1}, … ,

*Λ*

_{a}_{−1}∈

**F**

_{pm},

*Λ*

_{0}≠ 0,

*Λ*

_{1}≠ 0, are considered in detail. The structure, duals, Hamming and homogeneous distances of

*Λ*-constacyclic codes of length

*p*over ℛ

^{s}_{a}are established. It is shown that self-dual

*Λ*-constacyclic codes of length

*p*over ℛ

^{s}_{a}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 ℛ

_{a}for different units

*α*,

*β*.

#### Recommended Citation

Dinh, Hai Q.; Dhompongsa, Sompong; and Sriboonchitta, Songsak (2016). Repeated-Root Constacyclic Codes of Prime Power Length Over Fpm [u] / 〈ua〉 and Their Duals. *Discrete Mathematics* 339(6), 1706-1715. doi: 10.1016/J.DISC.2016.01.020 Retrieved from https://digitalcommons.kent.edu/mathpubs/22

#### Publisher

Elsevier

#### Publisher Location

Amsterdam, Netherlands