\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Trace description and Hamming weights of irreducible constacyclic codes

  • * Corresponding author: Anuradha Sharma.

    * Corresponding author: Anuradha Sharma. 
Abstract Full Text(HTML) Figure(0) / Table(2) Related Papers Cited by
  • 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].

    Mathematics Subject Classification: Primary: 94B15; Secondary: 11Txx.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Table 1.  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 $\mathbb{A}_0=1, \mathbb{A}_{9765000}=162760416, \mathbb{A}_{9766875}=81380208$
     | Show Table
    DownLoad: CSV

    Table 2.  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
     | Show Table
    DownLoad: CSV
  •   Y. Aubry and  P. LangevinOn the weights of binary irreducible cyclic codes, Proc. Workshop Coding Cryptogr., Norway, 2015. 
      L. D. Baumert  and  R. J. McEliece , Weights of irreducible cyclic codes, Inform. Control, 20 (1972) , 158-175. 
      L. D. Baumert  and  J. Mykkeltveit , Weight distributions of some irreducible cyclic codes, JPL Tech. Rep. 32-1526, 16 (1973) , 128-131. 
      B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, John Wiley & Sons Inc., New York, 1998.
      C. Ding , The weight distribution of some irreducible cyclic codes, IEEE Trans. Inf. Theory, 55 (2009) , 955-960. 
      C. Ding  and  J. Yang , Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013) , 434-446. 
      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. 
      M. Grassl, Code Tables: Bounds on the Parameters of Various Types of Codes, available at www.codetables.de
      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.
      S. J. Gurak , Period polynomials for $\mathbb F_q$ of fixed small degree, CRM Proc. Lect. Notes, 36 (2004) , 127-145. 
      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. 
      A. Hoshi , Explicit lifts of quantic Jacobi sums and period polynomials for $\mathbb F_q$, Proc. Japan Acad., 82 (2006) , 87-92. 
      P. Langevin, A new class of two weight codes, in Proc. 3rd Int. Conf. Finite Fields Appl., Cambridge Univ. Press, 1996, 181–187.
      F. MacWilliams  and  J. Seery , The weight distributions of some minimal cyclic codes, IEEE Trans. Inf. Theory, 27 (1981) , 796-806. 
      F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publ. Co., Amsterdam, 1977.
      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. 
      G. Myerson , Period polynomials and Gauss sums for finite fields, Acta Arith., 39 (1981) , 251-264. 
      B. Schmidt  and  C. White , All two-weight irreducible cyclic codes?, Finite Fields Appl., 8 (2002) , 1-17. 
      A. Sharma  and  G. K. Bakshi , The weight distribution of some irreducible cyclic codes, Finite Fields Appl., 18 (2012) , 144-159. 
      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. 
      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.
      Y. Song and Z. Li, The weight enumerator of some irreducible cyclic codes, preprint, arXiv: 1202.2907v1
      T. Storer, Cyclotomy and Difference Sets, Markham Publ. Company, Chicago, 1967.
      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.
      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. 
  • 加载中

Tables(2)

SHARE

Article Metrics

HTML views(367) PDF downloads(437) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return