Codes over $  \frak m $-adic completion rings

Saadoun Mahmoudi; Karim Samei

doi:10.3934/amc.2020122

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2022, Volume 16, Issue 3: 503-524. Doi: 10.3934/amc.2020122

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Codes over $ \frak m $-adic completion rings

Saadoun Mahmoudi and
Karim Samei^,

Department of Mathematics, Bu Ali Sina University, Hamedan, Iran

^*Corresponding author
^*Corresponding author

Received: April 2020

Revised: September 2020

Early access: December 25, 2020

Published online: December 25, 2020

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Abstract

The theory of linear codes over finite rings has been generalized to linear codes over infinite rings in two special cases; the ring of $ p $-adic integers and formal power series ring. These rings are examples of complete discrete valuation rings (CDVRs). In this paper, we generalize the theory of linear codes over the above two rings to linear codes over complete local principal ideal rings. In particular, we obtain the structure of linear and constacyclic codes over CDVRs. For this generalization, first we study linear codes over $ \hat{R}_{ \frak m} $, where $ R $ is a commutative Noetherian ring, $ \frak m = \langle \gamma\rangle $ is a maximal ideal of $ R $, and $ \hat{R}_{ \frak m} $ denotes the $ \frak m $-adic completion of $ R $. We call these codes, $ \frak m $-adic codes. Using the structure of $ \frak m $-adic codes, we present the structure of linear and constacyclic codes over complete local principal ideal rings.

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Mathematics Subject Classification: Primary: 11T71, 94B05; Secondary: , 13J10.

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