May 2018, 12(2): 415-428. doi: 10.3934/amc.2018025

On some classes of codes with a few weights

1. 

School of Computer Science & Technology, Beijing Institute of Technology, Beijing 100081, China

2. 

Department of Mathematics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China

* Corresponding author: Zhimin Sun

Received  June 2017 Revised  February 2018 Published  March 2018

Fund Project: This work was supported by The National Science Foundation of China (No. 11171366)

We generalize the code constructed recently by Wang et al, and obtain many classes of codes with a few weights. The weight distribution of these codes is completely determined, and the minimum distance of the duals of these codes is determined. We also show that some subclasses of the duals of these codes are optimal. Furthermore, some parameters of the generalized Hamming weight of these codes are calculated in certain cases.

Citation: Yiwei Liu, Zihui Liu. On some classes of codes with a few weights. Advances in Mathematics of Communications, 2018, 12 (2) : 415-428. doi: 10.3934/amc.2018025
References:
[1]

A. R. AndersonC. DingT. Helleseth and T. Kløve, How to build robust shared control systems, Des., Codes Cryptogr., 15 (1998), 111-124. doi: 10.1023/A:1026421315292.

[2]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102. doi: 10.1109/TIT.2005.847722.

[3]

K. Ding and C. Ding, Bianry linear codes with three weights, IEEE Commun. Lett., 18 (2014), 1879-1882.

[4]

C. DingT. HellesethT. Kløve and X. Wang, A general construction of authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235. doi: 10.1109/TIT.2007.896872.

[5]

G. D. Forney, Dimension/length profiles and trellis complexity of linear block codes, IEEE Trans. Inf. Theory, 40 (1994), 1741-1752. doi: 10.1109/18.340452.

[6]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.

[7]

R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997.

[8]

M. Moisio, Explicit evaluation of some exponential sums, Finite Fields Their Appl., 15 (2009), 644-651. doi: 10.1016/j.ffa.2009.05.005.

[9]

Q. WangK. Ding and R. Xue, Binary linear codes with two weights, IEEE Commun. Lett., 19 (2015), 1097-1100. doi: 10.1109/LCOMM.2015.2431253.

[10]

Q. WangK. DingD. D. Lin and R. Xue, A kind of three-weight linear codes, Cryptogr. Commun., 9 (2017), 315-322. doi: 10.1007/s12095-015-0180-3.

[11]

V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inf. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.

[12]

M. H. YangJ. LiK. Q. Feng and D. D. Lin, Generalized Hamming weights of irreducible cyclic codes, IEEE Trans. Inf. Theory, 61 (2015), 4905-4913. doi: 10.1109/TIT.2015.2444013.

[13]

J. Yuan and C. Ding, Secret sharing schemes from two-weight codes, Electronic Notes in Discrete Mathematics, 15 (2003), P232. doi: 10.1016/S1571-0653(04)00592-X.

[14]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212. doi: 10.1109/TIT.2005.860412.

show all references

References:
[1]

A. R. AndersonC. DingT. Helleseth and T. Kløve, How to build robust shared control systems, Des., Codes Cryptogr., 15 (1998), 111-124. doi: 10.1023/A:1026421315292.

[2]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102. doi: 10.1109/TIT.2005.847722.

[3]

K. Ding and C. Ding, Bianry linear codes with three weights, IEEE Commun. Lett., 18 (2014), 1879-1882.

[4]

C. DingT. HellesethT. Kløve and X. Wang, A general construction of authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235. doi: 10.1109/TIT.2007.896872.

[5]

G. D. Forney, Dimension/length profiles and trellis complexity of linear block codes, IEEE Trans. Inf. Theory, 40 (1994), 1741-1752. doi: 10.1109/18.340452.

[6]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.

[7]

R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997.

[8]

M. Moisio, Explicit evaluation of some exponential sums, Finite Fields Their Appl., 15 (2009), 644-651. doi: 10.1016/j.ffa.2009.05.005.

[9]

Q. WangK. Ding and R. Xue, Binary linear codes with two weights, IEEE Commun. Lett., 19 (2015), 1097-1100. doi: 10.1109/LCOMM.2015.2431253.

[10]

Q. WangK. DingD. D. Lin and R. Xue, A kind of three-weight linear codes, Cryptogr. Commun., 9 (2017), 315-322. doi: 10.1007/s12095-015-0180-3.

[11]

V. K. Wei, Generalized Hamming weights for linear codes, IEEE Trans. Inf. Theory, 37 (1991), 1412-1418. doi: 10.1109/18.133259.

[12]

M. H. YangJ. LiK. Q. Feng and D. D. Lin, Generalized Hamming weights of irreducible cyclic codes, IEEE Trans. Inf. Theory, 61 (2015), 4905-4913. doi: 10.1109/TIT.2015.2444013.

[13]

J. Yuan and C. Ding, Secret sharing schemes from two-weight codes, Electronic Notes in Discrete Mathematics, 15 (2003), P232. doi: 10.1016/S1571-0653(04)00592-X.

[14]

J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212. doi: 10.1109/TIT.2005.860412.

Table 1.  The weight distribution of the code $\mathcal{C}_{A}$ in Theorem 5
WeightMultiplicity
01
$\frac{q+\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$$\frac{q-1}{2}-\frac{q-1}{2r^{m}}S(a)$
$\frac{q-\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$$\frac{q-1}{2}+\frac{q-1}{2r^{m}}S(a)$
WeightMultiplicity
01
$\frac{q+\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$$\frac{q-1}{2}-\frac{q-1}{2r^{m}}S(a)$
$\frac{q-\sqrt{q}}{4}+\frac{q+\sqrt{q}}{4r^{m}}S(a)$$\frac{q-1}{2}+\frac{q-1}{2r^{m}}S(a)$
Table 2.  The weight distribution of the code $\mathcal{C}_{A}$ in Theorem 10
WeightMultiplicity
01
$\frac{q}{4}$$\frac{q-3}{2}+(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}\frac{q-1}{2r^{m}}S(a)$
$\frac{q+(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}2\sqrt{q}}{4}$$\frac{q-1}{2}-(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}\frac{q-1}{2r^{m}}S(a)$
$\frac{q+(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}\sqrt{q}}{4}-\frac{q+\sqrt{q}}{4r^{m}}S(a)$1
WeightMultiplicity
01
$\frac{q}{4}$$\frac{q-3}{2}+(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}\frac{q-1}{2r^{m}}S(a)$
$\frac{q+(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}2\sqrt{q}}{4}$$\frac{q-1}{2}-(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}\frac{q-1}{2r^{m}}S(a)$
$\frac{q+(-1)^{\textrm{wt}(\boldsymbol{c}^{(0)})}\sqrt{q}}{4}-\frac{q+\sqrt{q}}{4r^{m}}S(a)$1
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