-
Previous Article
On the spectrum for the genera of maximal curves over small fields
- AMC Home
- This Issue
-
Next Article
Generalized nonlinearity of $ S$-boxes
Trace description and Hamming weights of irreducible constacyclic codes
1. | Department of Mathematics, IIIT-Delhi, New Delhi, 110020, India |
2. | Department of Mathematics, IIT Delhi, New Delhi, 110016, India |
Irreducible constacyclic codes constitute an important family of error-correcting codesand have applications in space communications.In this paper, we provide a trace description of irreducible constacyclic codes of length $n$ over the finite field $\mathbb{F}_{q}$ of order $q,$ where $n$ is a positive integer and $q$ is a prime power coprime to $n.$ As an application, we determine Hamming weight distributions of some irreducible constacyclic codes of length $n$ over $\mathbb{F}_{q}.$ We also derive a weight-divisibility theorem for irreducible constacyclic codes, and obtain both lower and upper bounds on the non-zero Hamming weights in irreducible constacyclic codes. Besides illustrating our results with examples, we list some optimal irreducible constacyclic codes that attain the distance bounds given in Grassl's Table [
References:
[1] |
Y. Aubry and P. Langevin, On the weights of binary irreducible cyclic codes, Proc. Workshop Coding Cryptogr., Norway, 2015.
![]() |
[2] |
L. D. Baumert and R. J. McEliece,
Weights of irreducible cyclic codes, Inform. Control, 20 (1972), 158-175.
|
[3] |
L. D. Baumert and J. Mykkeltveit,
Weight distributions of some irreducible cyclic codes, JPL Tech. Rep. 32-1526, 16 (1973), 128-131.
|
[4] |
B. C. Berndt, R. J. Evans and K. S. Williams,
Gauss and Jacobi Sums,
John Wiley & Sons Inc., New York, 1998. |
[5] |
C. Ding,
The weight distribution of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009), 955-960.
|
[6] |
C. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.
|
[7] |
X. Dong and S. Yin,
The trace representation of $λ$-constacyclic codes over the finite field $\mathbb F_q$, J. Liaoning Normal Univ.(Nat. Sci. ed.), 33 (2010), 129-131.
|
[8] |
M. Grassl,
Code Tables: Bounds on the Parameters of Various Types of Codes,
available at www.codetables.de |
[9] |
P. Grover and A. K. Bhandari,
A note on the weight distribution of minimal constacyclic codes, Bull. Malaysian Math. Sci. Soc., 39 (2016), 689-697.
doi: 10.1007/s40840-015-0134-0. |
[10] |
S. J. Gurak,
Period polynomials for $\mathbb F_q$ of fixed small degree, CRM Proc. Lect. Notes, 36 (2004), 127-145.
|
[11] |
T. Helleseth, T. Kløve and J. Mykkeltveit,
The weight distribution of irreducible cyclic codes with block length $n_1((q^{\ell}-1)/N)$, Discrete Math., 18 (1977), 179-211.
|
[12] |
A. Hoshi,
Explicit lifts of quantic Jacobi sums and period polynomials for $\mathbb F_q$, Proc. Japan Acad., 82 (2006), 87-92.
|
[13] |
P. Langevin, A new class of two weight codes, in Proc. 3rd Int. Conf. Finite Fields Appl., Cambridge Univ. Press, 1996, 181–187. |
[14] |
F. MacWilliams and J. Seery,
The weight distributions of some minimal cyclic codes, IEEE Trans. Inf. Theory, 27 (1981), 796-806.
|
[15] |
F. J. MacWilliams and N. J. A. Sloane,
The Theory of Error-Correcting Codes,
North-Holland Publ. Co., Amsterdam, 1977. |
[16] |
M. J. Moisio and K. O. Väänänen,
Two recursive algorithms for computing the weight distribution of certain irreducible cyclic codes, IEEE Trans. Inf. Theory, 45 (1999), 1244-1249.
|
[17] |
G. Myerson,
Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981), 251-264.
|
[18] |
B. Schmidt and C. White,
All two-weight irreducible cyclic codes?, Finite Fields Appl., 8 (2002), 1-17.
|
[19] |
A. Sharma and G. K. Bakshi,
The weight distribution of some irreducible cyclic codes, Finite Fields Appl., 18 (2012), 144-159.
|
[20] |
A. Sharma, G. K. Bakshi and M. Raka,
The weight distributions of irreducible cyclic codes of length $2^m$, Finite Fields Appl., 13 (2007), 1086-1095.
|
[21] |
A. Sharma and A. K. Sharma,
A note on weight distributions of irreducible cyclic codes,
Discrete Math. Algor. Appl. 6 (2014), 17 pages.
doi: 10.1142/S1793830914500414. |
[22] |
Y. Song and Z. Li,
The weight enumerator of some irreducible cyclic codes,
preprint, arXiv: 1202.2907v1 |
[23] |
T. Storer,
Cyclotomy and Difference Sets,
Markham Publ. Company, Chicago, 1967. |
[24] |
C. Tang, Y. Qi, M. Xu, B. Wang and Y. Yang,
A note on weight distributions of irreducible cyclic codes,
in Proc. Int. Conf. Inform. Commun. Tech. (ICT) 2014.
doi: 10.1049/cp.2014.0606. |
[25] |
J. Yang and L. Xia,
Complete solving of explicit evaluation of Gauss sums in the index 2 case, Sci. China Math., 53 (2010), 2525-2542.
|
show all references
References:
[1] |
Y. Aubry and P. Langevin, On the weights of binary irreducible cyclic codes, Proc. Workshop Coding Cryptogr., Norway, 2015.
![]() |
[2] |
L. D. Baumert and R. J. McEliece,
Weights of irreducible cyclic codes, Inform. Control, 20 (1972), 158-175.
|
[3] |
L. D. Baumert and J. Mykkeltveit,
Weight distributions of some irreducible cyclic codes, JPL Tech. Rep. 32-1526, 16 (1973), 128-131.
|
[4] |
B. C. Berndt, R. J. Evans and K. S. Williams,
Gauss and Jacobi Sums,
John Wiley & Sons Inc., New York, 1998. |
[5] |
C. Ding,
The weight distribution of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009), 955-960.
|
[6] |
C. Ding and J. Yang,
Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.
|
[7] |
X. Dong and S. Yin,
The trace representation of $λ$-constacyclic codes over the finite field $\mathbb F_q$, J. Liaoning Normal Univ.(Nat. Sci. ed.), 33 (2010), 129-131.
|
[8] |
M. Grassl,
Code Tables: Bounds on the Parameters of Various Types of Codes,
available at www.codetables.de |
[9] |
P. Grover and A. K. Bhandari,
A note on the weight distribution of minimal constacyclic codes, Bull. Malaysian Math. Sci. Soc., 39 (2016), 689-697.
doi: 10.1007/s40840-015-0134-0. |
[10] |
S. J. Gurak,
Period polynomials for $\mathbb F_q$ of fixed small degree, CRM Proc. Lect. Notes, 36 (2004), 127-145.
|
[11] |
T. Helleseth, T. Kløve and J. Mykkeltveit,
The weight distribution of irreducible cyclic codes with block length $n_1((q^{\ell}-1)/N)$, Discrete Math., 18 (1977), 179-211.
|
[12] |
A. Hoshi,
Explicit lifts of quantic Jacobi sums and period polynomials for $\mathbb F_q$, Proc. Japan Acad., 82 (2006), 87-92.
|
[13] |
P. Langevin, A new class of two weight codes, in Proc. 3rd Int. Conf. Finite Fields Appl., Cambridge Univ. Press, 1996, 181–187. |
[14] |
F. MacWilliams and J. Seery,
The weight distributions of some minimal cyclic codes, IEEE Trans. Inf. Theory, 27 (1981), 796-806.
|
[15] |
F. J. MacWilliams and N. J. A. Sloane,
The Theory of Error-Correcting Codes,
North-Holland Publ. Co., Amsterdam, 1977. |
[16] |
M. J. Moisio and K. O. Väänänen,
Two recursive algorithms for computing the weight distribution of certain irreducible cyclic codes, IEEE Trans. Inf. Theory, 45 (1999), 1244-1249.
|
[17] |
G. Myerson,
Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981), 251-264.
|
[18] |
B. Schmidt and C. White,
All two-weight irreducible cyclic codes?, Finite Fields Appl., 8 (2002), 1-17.
|
[19] |
A. Sharma and G. K. Bakshi,
The weight distribution of some irreducible cyclic codes, Finite Fields Appl., 18 (2012), 144-159.
|
[20] |
A. Sharma, G. K. Bakshi and M. Raka,
The weight distributions of irreducible cyclic codes of length $2^m$, Finite Fields Appl., 13 (2007), 1086-1095.
|
[21] |
A. Sharma and A. K. Sharma,
A note on weight distributions of irreducible cyclic codes,
Discrete Math. Algor. Appl. 6 (2014), 17 pages.
doi: 10.1142/S1793830914500414. |
[22] |
Y. Song and Z. Li,
The weight enumerator of some irreducible cyclic codes,
preprint, arXiv: 1202.2907v1 |
[23] |
T. Storer,
Cyclotomy and Difference Sets,
Markham Publ. Company, Chicago, 1967. |
[24] |
C. Tang, Y. Qi, M. Xu, B. Wang and Y. Yang,
A note on weight distributions of irreducible cyclic codes,
in Proc. Int. Conf. Inform. Commun. Tech. (ICT) 2014.
doi: 10.1049/cp.2014.0606. |
[25] |
J. Yang and L. Xia,
Complete solving of explicit evaluation of Gauss sums in the index 2 case, Sci. China Math., 53 (2010), 2525-2542.
|
5 | 78 | 4 | 1 | 2 | |
7 | 6536 | 6 | 5 | 3 | |
13 | 33988780 | 8 | 2 | 4 | |
25 | 6781684 | 6 | 2 | 3 | |
4 | 21 | 3 | 2 | 1 | |
7 | 600 | 4 | 3 | 2 | |
3 | 20 | 4 | 1 | 2 | |
3 | 182 | 6 | 1 | 2 | |
3 | 1640 | 8 | 1 | 2 | |
3 | 14762 | 10 | 1 | 2 | |
4 | 29127 | 9 | 2 | 3 | |
4 | 21 | 3 | 2 | 1 | |
5 | 97656 | 8 | 2 | 2 | |
7 | 29412 | 6 | 3 | 2 | |
7 | 1441200 | 8 | 3 | 2 | |
7 | 23539604 | 10 | 1 | 2 | |
7 | 12 | 2 | 3 | 2 | |
7 | 57 | 3 | 0 | 3 | |
7 | 19 | 3 | 4 | 3 | |
7 | 57 | 3 | 5 | 1 | |
7 | 400 | 4 | 1 | 1 | |
7 | 19608 | 6 | 3 | 3 | |
7 | 13072 | 6 | 4 | 3 | |
9 | 820 | 4 | 2 | 2 | |
9 | 2690420 | 8 | 7 | 2 | |
9 | 1345210 | 8 | 3 | 4 | |
13 | 134078 | 6 | 1 | 3 | |
13 | 16994390 | 8 | 5 | 4 | |
13 | 1190 | 4 | 6 | 4 | |
16 | 273 | 3 | 4 | 1 | |
19 | 60 | 2 | 15 | 1 | |
25 | 651 | 3 | 16 | 1 | |
25 | 10172526 | 6 | 3 | 3 |
5 | 78 | 4 | 1 | 2 | |
7 | 6536 | 6 | 5 | 3 | |
13 | 33988780 | 8 | 2 | 4 | |
25 | 6781684 | 6 | 2 | 3 | |
4 | 21 | 3 | 2 | 1 | |
7 | 600 | 4 | 3 | 2 | |
3 | 20 | 4 | 1 | 2 | |
3 | 182 | 6 | 1 | 2 | |
3 | 1640 | 8 | 1 | 2 | |
3 | 14762 | 10 | 1 | 2 | |
4 | 29127 | 9 | 2 | 3 | |
4 | 21 | 3 | 2 | 1 | |
5 | 97656 | 8 | 2 | 2 | |
7 | 29412 | 6 | 3 | 2 | |
7 | 1441200 | 8 | 3 | 2 | |
7 | 23539604 | 10 | 1 | 2 | |
7 | 12 | 2 | 3 | 2 | |
7 | 57 | 3 | 0 | 3 | |
7 | 19 | 3 | 4 | 3 | |
7 | 57 | 3 | 5 | 1 | |
7 | 400 | 4 | 1 | 1 | |
7 | 19608 | 6 | 3 | 3 | |
7 | 13072 | 6 | 4 | 3 | |
9 | 820 | 4 | 2 | 2 | |
9 | 2690420 | 8 | 7 | 2 | |
9 | 1345210 | 8 | 3 | 4 | |
13 | 134078 | 6 | 1 | 3 | |
13 | 16994390 | 8 | 5 | 4 | |
13 | 1190 | 4 | 6 | 4 | |
16 | 273 | 3 | 4 | 1 | |
19 | 60 | 2 | 15 | 1 | |
25 | 651 | 3 | 16 | 1 | |
25 | 10172526 | 6 | 3 | 3 |
3 | 20 | 4 | 1 | 2 | 12 |
4 | 21 | 3 | 2 | 1 | 16 |
4 | 7 | 3 | 0 | 3 | 4 |
5 | 78 | 4 | 3 | 2 | 60 |
5 | 6 | 2 | 3 | 1 | 5 |
7 | 4 | 2 | 5 | 2 | 3 |
7 | 57 | 3 | 5 | 1 | 49 |
7 | 8 | 2 | 5 | 1 | 7 |
7 | 19 | 3 | 4 | 3 | 15 |
9 | 5 | 2 | 5 | 2 | 4 |
3 | 20 | 4 | 1 | 2 | 12 |
4 | 21 | 3 | 2 | 1 | 16 |
4 | 7 | 3 | 0 | 3 | 4 |
5 | 78 | 4 | 3 | 2 | 60 |
5 | 6 | 2 | 3 | 1 | 5 |
7 | 4 | 2 | 5 | 2 | 3 |
7 | 57 | 3 | 5 | 1 | 49 |
7 | 8 | 2 | 5 | 1 | 7 |
7 | 19 | 3 | 4 | 3 | 15 |
9 | 5 | 2 | 5 | 2 | 4 |
[1] |
Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049 |
[2] |
Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409 |
[3] |
Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039 |
[4] |
Pankaj Kumar, Monika Sangwan, Suresh Kumar Arora. The weight distributions of some irreducible cyclic codes of length $p^n$ and $2p^n$. Advances in Mathematics of Communications, 2015, 9 (3) : 277-289. doi: 10.3934/amc.2015.9.277 |
[5] |
Chengju Li, Qin Yue, Ziling Heng. Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates. Advances in Mathematics of Communications, 2015, 9 (3) : 341-352. doi: 10.3934/amc.2015.9.341 |
[6] |
Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017 |
[7] |
Gerardo Vega, Jesús E. Cuén-Ramos. The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes. Advances in Mathematics of Communications, 2020, 14 (3) : 525-533. doi: 10.3934/amc.2020059 |
[8] |
Tongjiang Yan, Yanyan Liu, Yuhua Sun. Cyclic codes from two-prime generalized cyclotomic sequences of order 6. Advances in Mathematics of Communications, 2016, 10 (4) : 707-723. doi: 10.3934/amc.2016036 |
[9] |
Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021002 |
[10] |
Lanqiang Li, Shixin Zhu, Li Liu. The weight distribution of a class of p-ary cyclic codes and their applications. Advances in Mathematics of Communications, 2019, 13 (1) : 137-156. doi: 10.3934/amc.2019008 |
[11] |
Alexander Schaub, Olivier Rioul, Jean-Luc Danger, Sylvain Guilley, Joseph Boutros. Challenge codes for physically unclonable functions with Gaussian delays: A maximum entropy problem. Advances in Mathematics of Communications, 2020, 14 (3) : 491-505. doi: 10.3934/amc.2020060 |
[12] |
Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265 |
[13] |
Tonghui Zhang, Hong Lu, Shudi Yang. Two-weight and three-weight linear codes constructed from Weil sums. Mathematical Foundations of Computing, 2022, 5 (2) : 129-144. doi: 10.3934/mfc.2021041 |
[14] |
David Keyes. $\mathbb F_p$-codes, theta functions and the Hamming weight MacWilliams identity. Advances in Mathematics of Communications, 2012, 6 (4) : 401-418. doi: 10.3934/amc.2012.6.401 |
[15] |
Joaquim Borges, Josep Rifà, Victor Zinoviev. Completely regular codes by concatenating Hamming codes. Advances in Mathematics of Communications, 2018, 12 (2) : 337-349. doi: 10.3934/amc.2018021 |
[16] |
Jinmei Fan, Yanhai Zhang. Optimal quinary negacyclic codes with minimum distance four. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021043 |
[17] |
Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006 |
[18] |
B. K. Dass, Namita Sharma, Rashmi Verma. Characterization of extended Hamming and Golay codes as perfect codes in poset block spaces. Advances in Mathematics of Communications, 2018, 12 (4) : 629-639. doi: 10.3934/amc.2018037 |
[19] |
Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028 |
[20] |
Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]