doi: 10.3934/amc.2022002
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Galois LCD codes over rings

Key Laboratory of Mathematical Theory and Computation in Information Security, Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received  March 2021 Revised  October 2021 Early access February 2022

Fund Project: The author is supported by The National Natural Science Foundation of China (No. 12071027) and Beijing Natural Science Foundation (No. 1222016)

We define the generalized Galois inner product for codes over Frobenius rings, and then present the Galois linear complementary dual (LCD) codes over such rings which generalize the Galois LCD codes over finite fields defined in a recent reference. We describe the judging criterions for a generalized Galois LCD code over a Frobenius ring by introducing the rank of a matrix over such a ring. We also present necessary and sufficient conditions for the generalized Galois LCD codes over chain rings. The structure of the generalized Galois LCD codes over chain rings can be used to construct Galois LCD codes over finite fields. By using the structure of the generalized Galois LCD codes, we also give the description of constacyclic LCD codes over a class of chain rings.

Citation: Zihui Liu. Galois LCD codes over rings. Advances in Mathematics of Communications, doi: 10.3934/amc.2022002
References:
[1]

Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$, Q. J. Math., 42 (1991), 387-391.  doi: 10.1093/qmath/42.1.387.

[2]

M. C. V. Amarra and F. R. Nemenzo, On $(1-u)$-cyclic codes over $F_{p^k}+uF_{p^k}$, Appl. Math. Letters, 21 (2008), 1129-1133.  doi: 10.1016/j.aml.2007.07.035.

[3]

Y. Cao, On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.  doi: 10.1016/j.ffa.2013.07.001.

[4]

C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun., 10 (2016), 131-150.  doi: 10.3934/amc.2016.10.131.

[5]

C. CarletS. MesnagerC. M. Tang and Y. F. Qi, On $\sigma$-LCD codes, IEEE Trans. Inf. Theory, 65 (2019), 1694-1704.  doi: 10.1109/TIT.2018.2873130.

[6]

C. CarletS. MesnagerC. M. TangY. F. Qi and R. Pellikaam, Linear codes over $F_q$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inf. Theory, 64 (2018), 3010-3017.  doi: 10.1109/TIT.2018.2789347.

[7]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3p^s$ and their duals, Discret. Math., 313 (2013), 983-991.  doi: 10.1016/j.disc.2013.01.024.

[8]

H. Q. DinhS. Dhompongsa and S. Sriboonchitta, Repeated-root constacyclic codes of prime power length over $F_{p^m}[u]/\langle u^a\rangle$ and their duals, Discret. Math., 339 (2016), 1706-1715.  doi: 10.1016/j.disc.2016.01.020.

[9]

H. Q. DinhB. T. Nguyen and S. Sriboonchitta, Constacyclic codes over finite commutative semi-simple rings, Finite Fields Appl., 45 (2017), 1-18.  doi: 10.1016/j.ffa.2016.11.008.

[10]

Y. Fan and L. Zhang, Galois self-dual constacyclic codes, Des. Codes Cryptogr., 84 (2017), 473-492.  doi: 10.1007/s10623-016-0282-8.

[11]

M. Hall, A type of algebraic colsure, Ann. of Math., 40 (1939), 360-369.  doi: 10.2307/1968924.

[12]

L. F. Jin, Construction of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63 (2017), 2843-2847. 

[13]

S. JitmanS. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63.  doi: 10.3934/amc.2012.6.39.

[14]

C. LiC. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inf. Theory, 63 (2017), 4344-4356.  doi: 10.1109/TIT.2017.2672961.

[15]

S. LiC. LiC. Ding and H. Liu, Two families of LCD BCH codes, IEEE Trans. Inf. Theory, 63 (2017), 5699-5717. 

[16]

X. S. LiuY. Fan and H. L. Liu, Galois LCD codes over finite fields, Finite Fields Appl., 49 (2018), 227-242.  doi: 10.1016/j.ffa.2017.10.001.

[17]

X. S. Liu and H. Liu, $\sigma$-LCD codes over finite chain rings, Des. Codes Cryptogr., 88 (2020), 727-746.  doi: 10.1007/s10623-019-00706-w.

[18]

X. S. Liu and H. L. Liu, LCD codes over finite chain rings, Finite Fields Appl., 34 (2015), 1-19.  doi: 10.1016/j.ffa.2015.01.004.

[19]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974.

[20]

B. R. McDonald, Linear Algebra Over Commutative Rings, Marcel Dekker, Inc., New York, 1984.

[21]

S. MesnagerC. Tang and Y. Qi, Complementary dual algebraic geometry codes, IEEE Trans. Inf. Theory, 64 (2018), 2390-2397.  doi: 10.1109/TIT.2017.2766075.

[22]

G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Eng. Commun. Comput., 10 (2000), 489-506.  doi: 10.1007/PL00012382.

[23]

A. Sharma and T. Sidana, On the structure and distances of repeated-root constacyclic codes of prime power lengths over finite commutative chain rings, IEEE Trans. Inf. Theory, 65 (2019), 1072-1084.  doi: 10.1109/TIT.2018.2864293.

[24]

M. J. ShiD. T. HuangL. Sok and P. Solé, Double circulant self-dual and LCD codes over Galois rings, Adv. Math. Commun., 13 (2019), 171-183.  doi: 10.3934/amc.2019011.

[25]

S. Szabo and J. A. Wood, Properties of dual codes defined by nondegenerate forms, J. Algebra Comb. Discrete Appl., 4 (2017), 105-113.  doi: 10.13069/jacodesmath.284934.

[26]

Z. X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/5350.

[27]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.  doi: 10.1353/ajm.1999.0024.

[28]

J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc., 136 (2008), 699-706.  doi: 10.1090/S0002-9939-07-09164-2.

[29]

X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discret. Math., 126 (1994), 391-393.  doi: 10.1016/0012-365X(94)90283-6.

show all references

References:
[1]

Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic $p$, Q. J. Math., 42 (1991), 387-391.  doi: 10.1093/qmath/42.1.387.

[2]

M. C. V. Amarra and F. R. Nemenzo, On $(1-u)$-cyclic codes over $F_{p^k}+uF_{p^k}$, Appl. Math. Letters, 21 (2008), 1129-1133.  doi: 10.1016/j.aml.2007.07.035.

[3]

Y. Cao, On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.  doi: 10.1016/j.ffa.2013.07.001.

[4]

C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun., 10 (2016), 131-150.  doi: 10.3934/amc.2016.10.131.

[5]

C. CarletS. MesnagerC. M. Tang and Y. F. Qi, On $\sigma$-LCD codes, IEEE Trans. Inf. Theory, 65 (2019), 1694-1704.  doi: 10.1109/TIT.2018.2873130.

[6]

C. CarletS. MesnagerC. M. TangY. F. Qi and R. Pellikaam, Linear codes over $F_q$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inf. Theory, 64 (2018), 3010-3017.  doi: 10.1109/TIT.2018.2789347.

[7]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3p^s$ and their duals, Discret. Math., 313 (2013), 983-991.  doi: 10.1016/j.disc.2013.01.024.

[8]

H. Q. DinhS. Dhompongsa and S. Sriboonchitta, Repeated-root constacyclic codes of prime power length over $F_{p^m}[u]/\langle u^a\rangle$ and their duals, Discret. Math., 339 (2016), 1706-1715.  doi: 10.1016/j.disc.2016.01.020.

[9]

H. Q. DinhB. T. Nguyen and S. Sriboonchitta, Constacyclic codes over finite commutative semi-simple rings, Finite Fields Appl., 45 (2017), 1-18.  doi: 10.1016/j.ffa.2016.11.008.

[10]

Y. Fan and L. Zhang, Galois self-dual constacyclic codes, Des. Codes Cryptogr., 84 (2017), 473-492.  doi: 10.1007/s10623-016-0282-8.

[11]

M. Hall, A type of algebraic colsure, Ann. of Math., 40 (1939), 360-369.  doi: 10.2307/1968924.

[12]

L. F. Jin, Construction of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63 (2017), 2843-2847. 

[13]

S. JitmanS. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63.  doi: 10.3934/amc.2012.6.39.

[14]

C. LiC. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inf. Theory, 63 (2017), 4344-4356.  doi: 10.1109/TIT.2017.2672961.

[15]

S. LiC. LiC. Ding and H. Liu, Two families of LCD BCH codes, IEEE Trans. Inf. Theory, 63 (2017), 5699-5717. 

[16]

X. S. LiuY. Fan and H. L. Liu, Galois LCD codes over finite fields, Finite Fields Appl., 49 (2018), 227-242.  doi: 10.1016/j.ffa.2017.10.001.

[17]

X. S. Liu and H. Liu, $\sigma$-LCD codes over finite chain rings, Des. Codes Cryptogr., 88 (2020), 727-746.  doi: 10.1007/s10623-019-00706-w.

[18]

X. S. Liu and H. L. Liu, LCD codes over finite chain rings, Finite Fields Appl., 34 (2015), 1-19.  doi: 10.1016/j.ffa.2015.01.004.

[19]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker, New York, 1974.

[20]

B. R. McDonald, Linear Algebra Over Commutative Rings, Marcel Dekker, Inc., New York, 1984.

[21]

S. MesnagerC. Tang and Y. Qi, Complementary dual algebraic geometry codes, IEEE Trans. Inf. Theory, 64 (2018), 2390-2397.  doi: 10.1109/TIT.2017.2766075.

[22]

G. H. Norton and A. Sălăgean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Eng. Commun. Comput., 10 (2000), 489-506.  doi: 10.1007/PL00012382.

[23]

A. Sharma and T. Sidana, On the structure and distances of repeated-root constacyclic codes of prime power lengths over finite commutative chain rings, IEEE Trans. Inf. Theory, 65 (2019), 1072-1084.  doi: 10.1109/TIT.2018.2864293.

[24]

M. J. ShiD. T. HuangL. Sok and P. Solé, Double circulant self-dual and LCD codes over Galois rings, Adv. Math. Commun., 13 (2019), 171-183.  doi: 10.3934/amc.2019011.

[25]

S. Szabo and J. A. Wood, Properties of dual codes defined by nondegenerate forms, J. Algebra Comb. Discrete Appl., 4 (2017), 105-113.  doi: 10.13069/jacodesmath.284934.

[26]

Z. X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/5350.

[27]

J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121 (1999), 555-575.  doi: 10.1353/ajm.1999.0024.

[28]

J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc., 136 (2008), 699-706.  doi: 10.1090/S0002-9939-07-09164-2.

[29]

X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discret. Math., 126 (1994), 391-393.  doi: 10.1016/0012-365X(94)90283-6.

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