# American Institute of Mathematical Sciences

February  2018, 12(1): 123-141. doi: 10.3934/amc.2018008

## 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

Received  October 2016 Published  March 2018

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 [8].

Citation: Anuradha Sharma, Saroj Rani. Trace description and Hamming weights of irreducible constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 123-141. doi: 10.3934/amc.2018008
##### References:
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##### References:
 [1] Y. Aubry and P. Langevin, On the weights of binary irreducible cyclic codes, Proc. Workshop Coding Cryptogr., Norway, 2015.   Google Scholar [2] L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Inform. Control, 20 (1972), 158-175.   Google Scholar [3] L. D. Baumert and J. Mykkeltveit, Weight distributions of some irreducible cyclic codes, JPL Tech. Rep. 32-1526, 16 (1973), 128-131.   Google Scholar [4] B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, John Wiley & Sons Inc., New York, 1998.  Google Scholar [5] C. Ding, The weight distribution of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009), 955-960.   Google Scholar [6] C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.   Google Scholar [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.   Google Scholar [8] M. Grassl, Code Tables: Bounds on the Parameters of Various Types of Codes, available at www.codetables.de Google Scholar [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.  Google Scholar [10] S. J. Gurak, Period polynomials for $\mathbb F_q$ of fixed small degree, CRM Proc. Lect. Notes, 36 (2004), 127-145.   Google Scholar [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.   Google Scholar [12] A. Hoshi, Explicit lifts of quantic Jacobi sums and period polynomials for $\mathbb F_q$, Proc. Japan Acad., 82 (2006), 87-92.   Google Scholar [13] P. Langevin, A new class of two weight codes, in Proc. 3rd Int. Conf. Finite Fields Appl., Cambridge Univ. Press, 1996, 181–187.  Google Scholar [14] F. MacWilliams and J. Seery, The weight distributions of some minimal cyclic codes, IEEE Trans. Inf. Theory, 27 (1981), 796-806.   Google Scholar [15] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publ. Co., Amsterdam, 1977.  Google Scholar [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.   Google Scholar [17] G. Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981), 251-264.   Google Scholar [18] B. Schmidt and C. White, All two-weight irreducible cyclic codes?, Finite Fields Appl., 8 (2002), 1-17.   Google Scholar [19] A. Sharma and G. K. Bakshi, The weight distribution of some irreducible cyclic codes, Finite Fields Appl., 18 (2012), 144-159.   Google Scholar [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.   Google Scholar [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.  Google Scholar [22] Y. Song and Z. Li, The weight enumerator of some irreducible cyclic codes, preprint, arXiv: 1202.2907v1 Google Scholar [23] T. Storer, Cyclotomy and Difference Sets, Markham Publ. Company, Chicago, 1967.  Google Scholar [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.  Google Scholar [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.   Google Scholar
Weight distribution of the $q$-ary code $\mathcal{M}_1^{(m,i)}$
 $\boldsymbol{q}$ $\boldsymbol{m}$ $\boldsymbol{k}$ $\boldsymbol{i}$ $\boldsymbol{L}$ $^1\textbf{Weight distribution of the$\boldsymbol{q}$-ary code$\mathcal{M}_1^{(m,i)}$}$ 5 78 4 1 2 $\mathbb{A}_0=1, \mathbb{A}_{60}=\mathbb{A}_{65}=312$ 7 6536 6 5 3 $\mathbb{A}_{0}=1, \mathbb{A}_{5572}=\mathbb{A}_{5628}=\mathbb{A}_{5607}=39216$ 13 33988780 8 2 4 $\mathbb{A}_{0}=1, \mathbb{A}_{31376904}=\mathbb{A}_{31373810}=\mathbb{A}_{31374720}$ $=\mathbb{A}_{31371600}=203932680$ 25 6781684 6 2 3 $\mathbb{A}_{0}=1, \mathbb{A}_{6510000}=162760416, \mathbb{A}_{6511250}=81380208$ 4 21 3 2 1 $\mathbb{A}_{0}=1, \mathbb{A}_{16}=63$ 7 600 4 3 2 $\mathbb{A}_0=1, \mathbb{A}_{504}=\mathbb{A}_{525}=1200$ 3 20 4 1 2 $\mathbb{A}_0=1, \mathbb{A}_{12}=40, \mathbb{A}_{15}=40$ 3 182 6 1 2 $\mathbb{A}_0=1, \mathbb{A}_{117}=364, \mathbb{A}_{126}=364$ 3 1640 8 1 2 $\mathbb{A}_0=1, \mathbb{A}_{1080}=3280, \mathbb{A}_{1107}=3280$ 3 14762 10 1 2 $\mathbb{A}_0=1, \mathbb{A}_{9801}=29524, \mathbb{A}_{9882}=29524$ 4 29127 9 2 3 $\mathbb{A}_0=1, \mathbb{A}_{21760}=87381, \mathbb{A}_{21888}=174762$ 4 21 3 2 1 $\mathbb{A}_0=1, \mathbb{A}_{16}=63$ 5 97656 8 2 2 $\mathbb{A}_0=1, \mathbb{A}_{78000}=195312, \mathbb{A}_{78250}=195312$ 7 29412 6 3 2 $\mathbb{A}_0=1, \mathbb{A}_{25137}=58824, \mathbb{A}_{25284}=58824$ 7 1441200 8 3 2 $\mathbb{A}_0=1, \mathbb{A}_{1234800}=2882400, \mathbb{A}_{1235829}=2882400$ 7 23539604 10 1 2 $\mathbb{A}_0=1, \mathbb{A}_{20175603}=141237624, \mathbb{A}_{20178004}=141237624$ 7 12 2 3 2 $\mathbb{A}_0=1, \mathbb{A}_{9}=24, \mathbb{A}_{12}=24$ 7 57 3 0 3 $\mathbb{A}_0=1, \mathbb{A}_{45}=114, \mathbb{A}_{48}=114, \mathbb{A}_{54}=114$ 7 19 3 4 3 $\mathbb{A}_0=1, \mathbb{A}_{15}=114, \mathbb{A}_{16}=114, \mathbb{A}_{18}=114$ 7 57 3 5 1 $\mathbb{A}_0=1, \mathbb{A}_{49}=342$ 7 400 4 1 1 $\mathbb{A}_0=1, \mathbb{A}_{343}=2400$ 7 19608 6 3 3 $\mathbb{A}_0=1, \mathbb{A}_{16716}=39216, \mathbb{A}_{16821}=39216,$ $\mathbb{A}_{16884}=39216$ 7 13072 6 4 3 $\mathbb{A}_0=1, \mathbb{A}_{11144}=39216, \mathbb{A}_{11256}=39216,$ $\mathbb{A}_{11214}=39216$ 9 820 4 2 2 $\mathbb{A}_0=1, \mathbb{A}_{720}=3280, \mathbb{A}_{738}=3280$ 9 2690420 8 7 2 $\mathbb{A}_0=1, \mathbb{A}_{2391120}=21523360, \mathbb{A}_{2391849}=21523360$ 9 1345210 8 3 4 $\mathbb{A}_0=1, \mathbb{A}_{1195560}=32285040, \mathbb{A}_{1196289}=10761680$ 13 134078 6 1 3 $\mathbb{A}_0=1, \mathbb{A}_{123669}=1608936, \mathbb{A}_{123760}=1608936,$ $\mathbb{A}_{123864}=1608936$ 13 16994390 8 5 4 $\mathbb{A}_0=1, \mathbb{A}_{15688452}=203932680, \mathbb{A}_{15686905}=203932680,$ $\mathbb{A}_{15687360}=203932680, \mathbb{A}_{15685800}=203932680$ 13 1190 4 6 4 $\mathbb{A}_0=1, \mathbb{A}_{1100}=7140, \mathbb{A}_{1104}=7140, \mathbb{A}_{1110}=7140,$ $\mathbb{A}_{1080}=7140$ 16 273 3 4 1 $\mathbb{A}_0=1, \mathbb{A}_{256}=4095$ 19 60 2 15 1 $\mathbb{A}_0=1, \mathbb{A}_{57}=360$ 25 651 3 16 1 $\mathbb{A}_0=1, \mathbb{A}_{625}=15624$ 25 10172526 6 3 3 $\mathbb{A}_0=1, \mathbb{A}_{9765000}=162760416, \mathbb{A}_{9766875}=81380208$
 $\boldsymbol{q}$ $\boldsymbol{m}$ $\boldsymbol{k}$ $\boldsymbol{i}$ $\boldsymbol{L}$ $^1\textbf{Weight distribution of the$\boldsymbol{q}$-ary code$\mathcal{M}_1^{(m,i)}$}$ 5 78 4 1 2 $\mathbb{A}_0=1, \mathbb{A}_{60}=\mathbb{A}_{65}=312$ 7 6536 6 5 3 $\mathbb{A}_{0}=1, \mathbb{A}_{5572}=\mathbb{A}_{5628}=\mathbb{A}_{5607}=39216$ 13 33988780 8 2 4 $\mathbb{A}_{0}=1, \mathbb{A}_{31376904}=\mathbb{A}_{31373810}=\mathbb{A}_{31374720}$ $=\mathbb{A}_{31371600}=203932680$ 25 6781684 6 2 3 $\mathbb{A}_{0}=1, \mathbb{A}_{6510000}=162760416, \mathbb{A}_{6511250}=81380208$ 4 21 3 2 1 $\mathbb{A}_{0}=1, \mathbb{A}_{16}=63$ 7 600 4 3 2 $\mathbb{A}_0=1, \mathbb{A}_{504}=\mathbb{A}_{525}=1200$ 3 20 4 1 2 $\mathbb{A}_0=1, \mathbb{A}_{12}=40, \mathbb{A}_{15}=40$ 3 182 6 1 2 $\mathbb{A}_0=1, \mathbb{A}_{117}=364, \mathbb{A}_{126}=364$ 3 1640 8 1 2 $\mathbb{A}_0=1, \mathbb{A}_{1080}=3280, \mathbb{A}_{1107}=3280$ 3 14762 10 1 2 $\mathbb{A}_0=1, \mathbb{A}_{9801}=29524, \mathbb{A}_{9882}=29524$ 4 29127 9 2 3 $\mathbb{A}_0=1, \mathbb{A}_{21760}=87381, \mathbb{A}_{21888}=174762$ 4 21 3 2 1 $\mathbb{A}_0=1, \mathbb{A}_{16}=63$ 5 97656 8 2 2 $\mathbb{A}_0=1, \mathbb{A}_{78000}=195312, \mathbb{A}_{78250}=195312$ 7 29412 6 3 2 $\mathbb{A}_0=1, \mathbb{A}_{25137}=58824, \mathbb{A}_{25284}=58824$ 7 1441200 8 3 2 $\mathbb{A}_0=1, \mathbb{A}_{1234800}=2882400, \mathbb{A}_{1235829}=2882400$ 7 23539604 10 1 2 $\mathbb{A}_0=1, \mathbb{A}_{20175603}=141237624, \mathbb{A}_{20178004}=141237624$ 7 12 2 3 2 $\mathbb{A}_0=1, \mathbb{A}_{9}=24, \mathbb{A}_{12}=24$ 7 57 3 0 3 $\mathbb{A}_0=1, \mathbb{A}_{45}=114, \mathbb{A}_{48}=114, \mathbb{A}_{54}=114$ 7 19 3 4 3 $\mathbb{A}_0=1, \mathbb{A}_{15}=114, \mathbb{A}_{16}=114, \mathbb{A}_{18}=114$ 7 57 3 5 1 $\mathbb{A}_0=1, \mathbb{A}_{49}=342$ 7 400 4 1 1 $\mathbb{A}_0=1, \mathbb{A}_{343}=2400$ 7 19608 6 3 3 $\mathbb{A}_0=1, \mathbb{A}_{16716}=39216, \mathbb{A}_{16821}=39216,$ $\mathbb{A}_{16884}=39216$ 7 13072 6 4 3 $\mathbb{A}_0=1, \mathbb{A}_{11144}=39216, \mathbb{A}_{11256}=39216,$ $\mathbb{A}_{11214}=39216$ 9 820 4 2 2 $\mathbb{A}_0=1, \mathbb{A}_{720}=3280, \mathbb{A}_{738}=3280$ 9 2690420 8 7 2 $\mathbb{A}_0=1, \mathbb{A}_{2391120}=21523360, \mathbb{A}_{2391849}=21523360$ 9 1345210 8 3 4 $\mathbb{A}_0=1, \mathbb{A}_{1195560}=32285040, \mathbb{A}_{1196289}=10761680$ 13 134078 6 1 3 $\mathbb{A}_0=1, \mathbb{A}_{123669}=1608936, \mathbb{A}_{123760}=1608936,$ $\mathbb{A}_{123864}=1608936$ 13 16994390 8 5 4 $\mathbb{A}_0=1, \mathbb{A}_{15688452}=203932680, \mathbb{A}_{15686905}=203932680,$ $\mathbb{A}_{15687360}=203932680, \mathbb{A}_{15685800}=203932680$ 13 1190 4 6 4 $\mathbb{A}_0=1, \mathbb{A}_{1100}=7140, \mathbb{A}_{1104}=7140, \mathbb{A}_{1110}=7140,$ $\mathbb{A}_{1080}=7140$ 16 273 3 4 1 $\mathbb{A}_0=1, \mathbb{A}_{256}=4095$ 19 60 2 15 1 $\mathbb{A}_0=1, \mathbb{A}_{57}=360$ 25 651 3 16 1 $\mathbb{A}_0=1, \mathbb{A}_{625}=15624$ 25 10172526 6 3 3 $\mathbb{A}_0=1, \mathbb{A}_{9765000}=162760416, \mathbb{A}_{9766875}=81380208$
Some examples of optimal irreducible constacyclic codes $\mathcal{M}_1^{(m,i)}$ over $\mathbb{F}_{q}$
 $\boldsymbol{q}$ $\boldsymbol{m}$ $\boldsymbol{k}$ $\boldsymbol{i}$ $\boldsymbol{L}$ $\boldsymbol{d}$ 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
 $\boldsymbol{q}$ $\boldsymbol{m}$ $\boldsymbol{k}$ $\boldsymbol{i}$ $\boldsymbol{L}$ $\boldsymbol{d}$ 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
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