Advances in Mathematics of Communications doi:10.3934/amc.2019019 Volume 13, No. 2, 2019, 281–296 SOME NEW CONSTRUCTIONS OF ISODUAL AND LCD CODES OVER FINITE FIELDS Fatma-Zohra Benahmed University M’Hamed Bougara of Boumerdes Faculty of Sciences, Boumerdes, Algeria Kenza Guenda and Aicha Batoul ∗ University of Science and Technology: Houari Boumediene Faculty of Mathematics, Algiers, 16411, Algeria Thomas Aaron Gulliver Department of Electrical and Computer Engineering University of Victoria Victoria, BC V8W 2Y2, Canada (Communicated by Sihem Mesnager) Abstract. This paper presents some new constructions of LCD, isodual, self- dual and LCD-isodual codes based on the structure of repeated-root consta- cyclic codes. We first characterize repeated-root constacyclic codes in terms of their generator polynomials and lengths. Then we provide simple conditions on the existence of repeated-root codes which are either self-dual negacyclic or LCD cyclic and negacyclic. This leads to the construction of LCD, self-dual, isodual, and LCD-isodual cyclic and negacyclic codes. Repeated-root constacyclic codes were first studied by Berman [6] and Falkner et al. [14]. Since then several authors have studied these codes. Dinh determined the generator polynomials of constacyclic codes over F q of lengths 3p r and 6p r in [12] and [13], respectively. These results have been extended to more general lengths by Bakshi [2] and Chen et al. [10] who gave the generator polynomials of all constacyclic codes of lengths 2 t p r and lp r with l prime, respectively. Linear code with complementary dual (LCD) were introduced by Massey [26, 23]. They provide an optimum linear coding solution for the two-user binary adder channel. Further, it has been shown that asymptotically good LCD codes exist [23]. Li et al. and Guenda [21, 16] investigated LCD BCH codes. Carlet et al. [9] gave a general construction of LCD codes from arbitrary linear codes. Isodual codes are codes which are equivalent to their dual. These codes are related to isodual lattices construction [1]. Recently, Batoul et al.[4],[5] constructed classes of isodual codes from an important class of linear codes, namely the class of constacyclic codes. In this paper, we construct new LCD, isodual and self-dual codes. Some of these codes are both isodual and LCD, and so are called LCD-isodual codes. These constructions are based on the structure of repeated-root constacyclic codes. We first characterize these codes in terms of their generator polynomials and lengths. 2010 Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases: Repeated-root constacyclic codes, negacyclic codes, self-dual codes, LCD codes, isodual codes, LCD-isodual codes. ∗ Corresponding author: Aicha Batoul. 281 c 2019 AIMS