On the construction of LCD codes over certain finite rings

• Main Author: Lina, Eusebio R., Jr.
• Format: text
• Language: English
• Published: Animo Repository 2016
• Subjects: Algebras; Linear; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_doctoral/482
• Institution: De La Salle University

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.