LCD codes and LCP of codes from units of group rings

Year 2019, , 486 - 492, 01.06.2019

Mehmet Emin Köroğlu

https://doi.org/10.16984/saufenbilder.459123

Abstract

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

References

• [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.