May  2019, 13(2): 267-280. doi: 10.3934/amc.2019018

Some classes of LCD codes and self-orthogonal codes over finite fields

1. 

School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

2. 

State Key Laboratory of Cryptology, P. O. Box 5159, Beijing 100878, China

3. 

School of Mathematics and Information, China West Normal University, Nanchong 637002, China

* Corresponding author: Chunming Tang

Received  March 2018 Revised  September 2018 Published  February 2019

Due to their important applications in theory and practice, linear complementary dual (LCD) codes and self-orthogonal codes have received much attention in the last decade. The objective of this paper is to extend a recent construction of binary LCD codes and self-orthogonal codes to the general $ p $-ary case, where $ p $ is an odd prime. Based on the extended construction, several classes of $ p $-ary linear codes are obtained. The characterizations of these linear codes to be LCD or self-orthogonal are derived. The duals of these linear codes are also studied. It turns out that the proposed linear codes are optimal in many cases in the sense that their parameters meet certain bounds on linear codes. The weight distributions of these linear codes are settled.

Citation: Xia Li, Feng Cheng, Chunming Tang, Zhengchun Zhou. Some classes of LCD codes and self-orthogonal codes over finite fields. Advances in Mathematics of Communications, 2019, 13 (2) : 267-280. doi: 10.3934/amc.2019018
References:
[1]

E. R. Berlekamp, Algebraic Coding Theory, Revised edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9407. Google Scholar

[2]

J. BringerC. CarletH. ChabanneS. Guilley and H. Maghrebi, Orthogonal direct sum masking, a smartcard friendly computation paradigm in a code, with Builtin protection against side-channel and fault attacks, Proceedings of WISTP 2014, Lecture Notes in Computer Science, 8501 (2014), 40-56. Google Scholar

[3]

A. K. CalderbankE. M. RainsP. W. Shor and N. J. A. Sloane, Quantum error correction and orthogonal geometry, Phys. Rev. Lett., 78 (1997), 405-408. doi: 10.1103/PhysRevLett.78.405. Google Scholar

[4]

A. R. CalderbankE. M. RainsP. W. Shor and N. J. A. Sloane, Quantum error correction via codes over $\mathbb{GF} (4)$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387. doi: 10.1109/18.681315. Google Scholar

[5]

C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, E. R. Pinto et al. (eds.), Coding Theory and Applications, CIM Series in Mathematical Sciences, 3 (2014), 97-105. Google Scholar

[6]

C. CarletX. ZengC. Li and L. Hu, Further properties of several classes of boolean functions with optimum algebraic immunity, Des. Codes Cryptogr, 52 (2009), 303-338. doi: 10.1007/s10623-009-9284-0. Google Scholar

[7]

C. CarletS. MesnagerC. TangY. Qi and R. Pellikaan, Linear codes over $\mathbb F_q$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inform. Theory, 64 (2018), 3010-3017. doi: 10.1109/TIT.2018.2789347. Google Scholar

[8]

C. CarletS. MesnagerC. Tang and Y. Qi, Euclidean and Hermitian LCD MDS codes, Designs, Codes Cryptogr., 86 (2018), 2605-2618. doi: 10.1007/s10623-018-0463-8. Google Scholar

[9]

C. Carlet, S. Mesnager, C. Tang and Y. Qi, On $\sigma$-LCD codes, IEEE Trans. Inform. Theory, accepted for publication. doi: 10.1109/TIT.2018.2873130. Google Scholar

[10]

H. ChenS. Ling and C. Xing, Quantum codes from concatenated algebraic-geometric codes, IEEE Trans. Inform. Theory, 51 (2005), 2915-2920. doi: 10.1109/TIT.2005.851760. Google Scholar

[11]

D. K. DalaiS. Maitra and S. Sarkar, Basic theory in construction of boolean functions with maximum possible annihilator immunity, Des. Codes Cryptogr, 40 (2006), 41-58. doi: 10.1007/s10623-005-6300-x. Google Scholar

[12]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inform. Theory, 61 (2015), 3265-3275. doi: 10.1109/TIT.2015.2420118. Google Scholar

[13]

S. T. DoughertyJ.-L. KimB. ÖzkayaL. Sok and P. Solé, The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices, International Journal of Information and Coding Theory, 4 (2017), 116-128. doi: 10.1504/IJICOT.2017.083827. Google Scholar

[14]

M. Esmaeili and S. Yari, On complementary-dual quasi-cyclic codes, Finite Fields Appl., 15 (2009), 375-386. doi: 10.1016/j.ffa.2009.01.002. Google Scholar

[15]

D. Gottesman, A class of quantum error-correcting codes saturating the quantum Hamming bounds, Phys. Rev. A, 54 (1996), 1862-1868. doi: 10.1103/PhysRevA.54.1862. Google Scholar

[16]

Z. HengC. Ding and Z. Zhou, Minimal linear codes over finite fields, Finite Fields and Their Applicaitons, 54 (2018), 176-196. doi: 10.1016/j.ffa.2018.08.010. Google Scholar

[17] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge university press, 2003. doi: 10.1017/CBO9780511807077. Google Scholar
[18]

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

[19]

L. Jin and C. Xing, Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes, IEEE Trans. Inform. Theory, 58 (2012), 5484-5489. doi: 10.1109/TIT.2011.2177066. Google Scholar

[20]

T. Kløve, Codes for Error Detection, World Scientific, 2007. doi: 10.1142/9789812770516. Google Scholar

[21]

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

[22]

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

[23]

C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr., 86 (2018), 2261-2278. doi: 10.1007/s10623-017-0447-0. Google Scholar

[24]

S. Lloyd, Binary block coding, Bell Labs Technical Journal, 36 (1957), 517-535. doi: 10.1002/j.1538-7305.1957.tb02410.x. Google Scholar

[25]

E. Lucas, Sur les congruences des nombres euleriennes et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. (France), 6 (1878), 49-54. Google Scholar

[26]

J. L. Massey, Reversible codes, Information and Control, 7 (1964), 369-380. doi: 10.1016/S0019-9958(64)90438-3. Google Scholar

[27]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342. doi: 10.1016/0012-365X(92)90563-U. Google Scholar

[28]

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

[29]

N. Sendrier, Linear codes with complementary duals meet the Gilbert–Varshamov bound, Discrete Math., 285 (2004), 345-347. doi: 10.1016/j.disc.2004.05.005. Google Scholar

[30]

A. Sharma, Self-dual and self-orthogonal negacyclic codes of length $2^mp^n$ over a finite field, Discrete Mathematics, 338 (2015), 576-592. doi: 10.1016/j.disc.2014.11.008. Google Scholar

[31]

X. ShiQ. Yue and S. Yang, New LCD MDS codes constructed from generalized Reed–Solomon codes, Cryptogr. Commun., 10 (2018), 1165-1182. doi: 10.1007/s12095-017-0274-1. Google Scholar

[32]

R. Townsend and E. Weldon, Self-orthogonal quasi-cyclic codes, IEEE Transactions on Information Theory, 13 (1967), 183-195. doi: 10.1109/TIT.1967.1053974. Google Scholar

[33]

K. K. Tzeng and C. R. P. Hartmann, On the minimum distance of certain reversible cyclic codes, IEEE Trans. Inform. Theory, 16 (1970), 644-646. doi: 10.1109/tit.1970.1054517. Google Scholar

[34]

H. YanH. LiuC. Li and S. Yang, Parameters of LCD BCH codes with two lengths, Advances in Mathematics of Communications, 12 (2018), 579-594. doi: 10.3934/amc.2018034. Google Scholar

[35]

T. Zhang and G. Ge, Quantum codes derived from certain classes of polynomials, IEEE Transactions on Information Theory, 62 (2016), 6638-6643. doi: 10.1109/TIT.2016.2612578. Google Scholar

[36]

Z. ZhouC. TangX. Li and C. Ding, Binary LCD Codes and Self-orthogonal Codes from a Generic Construction, IEEE Trans. Inform. Theory, 65 (2019), 16-27. doi: 10.1109/TIT.2018.2823704. Google Scholar

show all references

References:
[1]

E. R. Berlekamp, Algebraic Coding Theory, Revised edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9407. Google Scholar

[2]

J. BringerC. CarletH. ChabanneS. Guilley and H. Maghrebi, Orthogonal direct sum masking, a smartcard friendly computation paradigm in a code, with Builtin protection against side-channel and fault attacks, Proceedings of WISTP 2014, Lecture Notes in Computer Science, 8501 (2014), 40-56. Google Scholar

[3]

A. K. CalderbankE. M. RainsP. W. Shor and N. J. A. Sloane, Quantum error correction and orthogonal geometry, Phys. Rev. Lett., 78 (1997), 405-408. doi: 10.1103/PhysRevLett.78.405. Google Scholar

[4]

A. R. CalderbankE. M. RainsP. W. Shor and N. J. A. Sloane, Quantum error correction via codes over $\mathbb{GF} (4)$, IEEE Trans. Inform. Theory, 44 (1998), 1369-1387. doi: 10.1109/18.681315. Google Scholar

[5]

C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, E. R. Pinto et al. (eds.), Coding Theory and Applications, CIM Series in Mathematical Sciences, 3 (2014), 97-105. Google Scholar

[6]

C. CarletX. ZengC. Li and L. Hu, Further properties of several classes of boolean functions with optimum algebraic immunity, Des. Codes Cryptogr, 52 (2009), 303-338. doi: 10.1007/s10623-009-9284-0. Google Scholar

[7]

C. CarletS. MesnagerC. TangY. Qi and R. Pellikaan, Linear codes over $\mathbb F_q$ are equivalent to LCD codes for $q>3$, IEEE Trans. Inform. Theory, 64 (2018), 3010-3017. doi: 10.1109/TIT.2018.2789347. Google Scholar

[8]

C. CarletS. MesnagerC. Tang and Y. Qi, Euclidean and Hermitian LCD MDS codes, Designs, Codes Cryptogr., 86 (2018), 2605-2618. doi: 10.1007/s10623-018-0463-8. Google Scholar

[9]

C. Carlet, S. Mesnager, C. Tang and Y. Qi, On $\sigma$-LCD codes, IEEE Trans. Inform. Theory, accepted for publication. doi: 10.1109/TIT.2018.2873130. Google Scholar

[10]

H. ChenS. Ling and C. Xing, Quantum codes from concatenated algebraic-geometric codes, IEEE Trans. Inform. Theory, 51 (2005), 2915-2920. doi: 10.1109/TIT.2005.851760. Google Scholar

[11]

D. K. DalaiS. Maitra and S. Sarkar, Basic theory in construction of boolean functions with maximum possible annihilator immunity, Des. Codes Cryptogr, 40 (2006), 41-58. doi: 10.1007/s10623-005-6300-x. Google Scholar

[12]

C. Ding, Linear codes from some 2-designs, IEEE Trans. Inform. Theory, 61 (2015), 3265-3275. doi: 10.1109/TIT.2015.2420118. Google Scholar

[13]

S. T. DoughertyJ.-L. KimB. ÖzkayaL. Sok and P. Solé, The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices, International Journal of Information and Coding Theory, 4 (2017), 116-128. doi: 10.1504/IJICOT.2017.083827. Google Scholar

[14]

M. Esmaeili and S. Yari, On complementary-dual quasi-cyclic codes, Finite Fields Appl., 15 (2009), 375-386. doi: 10.1016/j.ffa.2009.01.002. Google Scholar

[15]

D. Gottesman, A class of quantum error-correcting codes saturating the quantum Hamming bounds, Phys. Rev. A, 54 (1996), 1862-1868. doi: 10.1103/PhysRevA.54.1862. Google Scholar

[16]

Z. HengC. Ding and Z. Zhou, Minimal linear codes over finite fields, Finite Fields and Their Applicaitons, 54 (2018), 176-196. doi: 10.1016/j.ffa.2018.08.010. Google Scholar

[17] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge university press, 2003. doi: 10.1017/CBO9780511807077. Google Scholar
[18]

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

[19]

L. Jin and C. Xing, Euclidean and Hermitian self-orthogonal algebraic geometry codes and their application to quantum codes, IEEE Trans. Inform. Theory, 58 (2012), 5484-5489. doi: 10.1109/TIT.2011.2177066. Google Scholar

[20]

T. Kløve, Codes for Error Detection, World Scientific, 2007. doi: 10.1142/9789812770516. Google Scholar

[21]

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

[22]

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

[23]

C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr., 86 (2018), 2261-2278. doi: 10.1007/s10623-017-0447-0. Google Scholar

[24]

S. Lloyd, Binary block coding, Bell Labs Technical Journal, 36 (1957), 517-535. doi: 10.1002/j.1538-7305.1957.tb02410.x. Google Scholar

[25]

E. Lucas, Sur les congruences des nombres euleriennes et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. (France), 6 (1878), 49-54. Google Scholar

[26]

J. L. Massey, Reversible codes, Information and Control, 7 (1964), 369-380. doi: 10.1016/S0019-9958(64)90438-3. Google Scholar

[27]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342. doi: 10.1016/0012-365X(92)90563-U. Google Scholar

[28]

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

[29]

N. Sendrier, Linear codes with complementary duals meet the Gilbert–Varshamov bound, Discrete Math., 285 (2004), 345-347. doi: 10.1016/j.disc.2004.05.005. Google Scholar

[30]

A. Sharma, Self-dual and self-orthogonal negacyclic codes of length $2^mp^n$ over a finite field, Discrete Mathematics, 338 (2015), 576-592. doi: 10.1016/j.disc.2014.11.008. Google Scholar

[31]

X. ShiQ. Yue and S. Yang, New LCD MDS codes constructed from generalized Reed–Solomon codes, Cryptogr. Commun., 10 (2018), 1165-1182. doi: 10.1007/s12095-017-0274-1. Google Scholar

[32]

R. Townsend and E. Weldon, Self-orthogonal quasi-cyclic codes, IEEE Transactions on Information Theory, 13 (1967), 183-195. doi: 10.1109/TIT.1967.1053974. Google Scholar

[33]

K. K. Tzeng and C. R. P. Hartmann, On the minimum distance of certain reversible cyclic codes, IEEE Trans. Inform. Theory, 16 (1970), 644-646. doi: 10.1109/tit.1970.1054517. Google Scholar

[34]

H. YanH. LiuC. Li and S. Yang, Parameters of LCD BCH codes with two lengths, Advances in Mathematics of Communications, 12 (2018), 579-594. doi: 10.3934/amc.2018034. Google Scholar

[35]

T. Zhang and G. Ge, Quantum codes derived from certain classes of polynomials, IEEE Transactions on Information Theory, 62 (2016), 6638-6643. doi: 10.1109/TIT.2016.2612578. Google Scholar

[36]

Z. ZhouC. TangX. Li and C. Ding, Binary LCD Codes and Self-orthogonal Codes from a Generic Construction, IEEE Trans. Inform. Theory, 65 (2019), 16-27. doi: 10.1109/TIT.2018.2823704. Google Scholar

Table 1.  The weight distribution of $ {\mathcal C}_{D_{t}} $ for $ 1\leq t\leq m $
Weight Multiplicity
$ 0 $ $ 1 $ time
$ \frac{(p-1)n-(p-1)K_{t}(k,m)}{p} $ for $ k=1,2,\cdots,m-1 $ $ (p-1)^k{m\choose k} $ times
$ \frac{(p-1){m\choose t}((p-1)^t+(-1)^{t+1})}{p} $ $ (p-1)^m $ times
Weight Multiplicity
$ 0 $ $ 1 $ time
$ \frac{(p-1)n-(p-1)K_{t}(k,m)}{p} $ for $ k=1,2,\cdots,m-1 $ $ (p-1)^k{m\choose k} $ times
$ \frac{(p-1){m\choose t}((p-1)^t+(-1)^{t+1})}{p} $ $ (p-1)^m $ times
Table 2.  The weight distribution of $ {\mathcal C}_{D_{t}} $ for $ 1\leq t\leq m $
Weight Multiplicity
$ 0 $ $ 1 $ time
$ \frac{\sum^t\limits_{i=1}(p-1)^{i+1}{{m}\choose{i}}-(p-1)^{t+1}{{m-1}\choose{t}}+p-1}{p} $ $ (p-1)m $ times
$ \frac{(p-1)n-(p-1)K_{t}(k-1,m-1)+(p-1)}{p} $ $ k=2,\cdots,m $ $ (p-1)^k{m\choose k} $ times
Weight Multiplicity
$ 0 $ $ 1 $ time
$ \frac{\sum^t\limits_{i=1}(p-1)^{i+1}{{m}\choose{i}}-(p-1)^{t+1}{{m-1}\choose{t}}+p-1}{p} $ $ (p-1)m $ times
$ \frac{(p-1)n-(p-1)K_{t}(k-1,m-1)+(p-1)}{p} $ $ k=2,\cdots,m $ $ (p-1)^k{m\choose k} $ times
Table 3.  The weight distribution of $ \mathcal C_{\overline{D}_ t} $ for $ 1\leq t\leq m $
Weight Multiplicity
$ 0 $ $ 1 $ time
$ \frac{(p-1)n-(p-1) \left ( K_{t}(k,m)+ K_{m}(k,m) \right )}{p} $ for $ k=1,2,\cdots,m-1 $ $ (p-1)^k{m\choose k} $ times
$ \frac{(p-1){m\choose t}((p-1)^t+(-1)^{t+1})+(p-1)((p-1)^m-(-1)^m)}{p} $ $ (p-1)^m $ times
Weight Multiplicity
$ 0 $ $ 1 $ time
$ \frac{(p-1)n-(p-1) \left ( K_{t}(k,m)+ K_{m}(k,m) \right )}{p} $ for $ k=1,2,\cdots,m-1 $ $ (p-1)^k{m\choose k} $ times
$ \frac{(p-1){m\choose t}((p-1)^t+(-1)^{t+1})+(p-1)((p-1)^m-(-1)^m)}{p} $ $ (p-1)^m $ times
Table 4.  The weight distribution of $ {\mathcal C}_{\overline{D}_{t}} $ for $ 1\leq t\leq m $
Weight Multiplicity
$ 0 $ $ 1 $ time
$ \frac{p(p-1)^{m}+\sum^t\limits_{i=1}(p-1)^{i+1}{{m}\choose{i}}-(p-1)^{t+1}{{m-1}\choose{t}}+p-1}{p} $ $ (p-1)m $ times
$ \frac{(p-1)n-(p-1)\left ( K_{t}(k-1,m-1)+K_{m}(k,m) -1\right )}{p} $ $ k=2,\cdots,m $ $ (p-1)^k{m\choose k} $ times
Weight Multiplicity
$ 0 $ $ 1 $ time
$ \frac{p(p-1)^{m}+\sum^t\limits_{i=1}(p-1)^{i+1}{{m}\choose{i}}-(p-1)^{t+1}{{m-1}\choose{t}}+p-1}{p} $ $ (p-1)m $ times
$ \frac{(p-1)n-(p-1)\left ( K_{t}(k-1,m-1)+K_{m}(k,m) -1\right )}{p} $ $ k=2,\cdots,m $ $ (p-1)^k{m\choose k} $ times
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