saujs Sakarya University Journal of Science 2147-835X Sakarya University 10.16984/saufenbilder.459123 Mathematical Sciences Matematik LCD codes and LCP of codes from units of group rings LCD codes and LCP of codes from units of group rings https://orcid.org/0000-0002-9173-4944 Köroğlu Mehmet Emin YILDIZ TEKNİK ÜNİVERSİTESİ 06 01 2019 23 3 486 492 09 11 2018 02 07 2019 Copyright © 1997, Sakarya University Journal of Science 1997 Sakarya University Journal of Science

A linear code with complementary dual(LCD) is a linear code such that LCD codes are of greatimportance due to their wide range of applications in consumer electronics,storage systems and cryptography. Group rings have a rich source of units. Alsothe well-known structural linear codes such as cyclic codes are within thefamily of group ring codes. Thus, group rings offer an affluent source forstructural codes that may lead to linear codes with good properties. In thiswork, we derive a condition for codes obtained from units of group rings to beLCD. We show that a special decomposition of group rings meet the LCDcondition. We also proposed a consruction of linear complementary pair (LCP) ofcodes.

A linear code with complementary dual(LCD) is a linear code such that LCD codes are of greatimportance due to their wide range of applications in consumer electronics,storage systems and cryptography. Group rings have a rich source of units. Alsothe well-known structural linear codes such as cyclic codes are within thefamily of group ring codes. Thus, group rings offer an affluent source forstructural codes that may lead to linear codes with good properties. In thiswork, we derive a condition for codes obtained from units of group rings to beLCD. We show that a special decomposition of group rings meet the LCDcondition. We also proposed a consruction of linear complementary pair (LCP) ofcodes.

 group rings unit derived codes LCD codes [1] J. L. Massey, “Linear codes with complementary duals,” Discrete Math, vol. 106, pp. 337–342, 1992.[2] J. L. Massey, “Reversible codes,” Information and Control, vol. 7, pp. 369–380, 1964.[3] N. Sendrier, “Linear codes with complementary duals meet the Gilbert-Varshamov bound,” Discrete Math, vol. 285, pp. 345-347, 2004.[4] C. P. Milies, K.S. Sudarshan, “An introduction to group rings,” vol.1. Springer, Nedherlands, 2002.[5] P. Hurley, T. Hurley, “Codes from zero-divisors and units in group rings,” Int. J. Inf. Coding Theory vol. 1 pp. 57-87, 2009.