Title

On the construction of LCD codes over certain finite rings

Date of Publication

2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Subject Categories

Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics Department

Thesis Adviser

Ederlina G. Nocon

Defense Panel Chair

Arlene A. Pascasio

Defense Panel Member

Isagani B. Jos
Fidel R. Nemenzo
Edmundo R. Perez, Jr.
Evangeeine P. Bautista

Abstract/Summary

Linear codes with complementary duals (LCD codes) are linear codes that intersect with their duals trivially. This paper presents some construction of LCD codes over finite fields applying Massey's characterization of LCD codes. We construct some classes of binary LCD codes using the permutation matrix and the all one matrix. Explicit construction of generator matrices of LCD codes using the generator matrices of self-dual codes and binary Hamming codes are given. We also revisit some known methods of combining two or more codes such us direct product, direct sum and Plotkin sum and determine whether such methods when applied to LCD codes will give rise to new LCD codes.
This paper also examines LCD codes over the nite non-chain rings R2 = F2 + vF2 + v2F2 and Rp = Fp + vFp + v2Fp, where v3 = v and p is an odd prime. We construct LCD codes over F2 and Fp as Gray images of LCD codes over R2 and Rp, respectively. In addition, we give necessary and su cient conditions for linear codes over R2 and Rp to be LCD.
Finally, we examine the LCD-ness of skew cyclic codes. Let Fq be a finite field of order q and be an automorphism on Fq. A skew cyclic code over Fq is a linear code C with the property that if (a0 a1 : : : an{u100000}1) 2 C, then ( (an{u100000}1) (a0) : : : (an{u100000}2)) 2 C. In this study, we give some conditions for a skew cyclic code to have a complementary viii dual. To this end, we revisit the properties of a noncommutative skew polynomial ring Fq[x ] of automorphism type and examine the algebraic structure of skew cyclic code using its skew polynomial representation. Using the result that skew cyclic codes are left ideals of the ring Fq[x ]=hxn{u100000}1i, we derive a characterization of a skew cyclic LCD code of length n.

Abstract Format

html

Language

English

Format

Electronic

Accession Number

CDTG006811

Shelf Location

Archives, The Learning Commons, 12F Henry Sy Sr. Hall

Physical Description

1 computer optical disc; 4 3/4 in

Keywords

Algebras; Linear

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