Advanced Search
Article Contents
Article Contents

The weight distribution of quasi-quadratic residue codes

The first author is supported by Simons Foundation grant MSN179747

Abstract Full Text(HTML) Figure(1) / Table(4) Related Papers Cited by
  • We investigate a family of codes called quasi-quadratic residue (QQR) codes. We are interested in these codes mainly for two reasons: Firstly, they have close relations with hyperelliptic curves and Goppa's Conjecture, and serve as a strong tool in studying those objects. Secondly, they are very good codes. Computational results show they have large minimum distances when $p\equiv 3 \pmod 8$.

    Our studies focus on the weight distributions of these codes. We will prove a new discovery about their weight polynomials, i.e. they are divisible by $(x^2 + y^2)^{d-1}$, where $d$ is the corresponding minimum distance. We also show that the weight distributions of these codes are asymptotically normal. Based on the relation between QQR codes and hyperelliptic curves, we will also prove a result on the point distribution on hyperelliptic curves.

    Mathematics Subject Classification: Primary: 94B15, 94B60; Secondary: 11G20.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  distribution comparison

    Table 1.  Computational Results

    $p$ $d$ $\delta$Divisible by
    320.33 $(x^2 + y^2)^3$
    1160.27 $(x^2 + y^2)^7$
    1980.21 $(x^2 + y^2)^7$
    43140.16 $(x^2 + y^2)^{15}$
    59180.15 $(x^2 + y^2)^{19}$
    67220.16 $(x^2 + y^2)^{23}$
     | Show Table
    DownLoad: CSV

    Table 2.  Weight polynomials posted on [18]

    $p$ $k$ $d$Divisible by
    894517 $(x+y)^{17}$
    974915 $(x+y)^{15}$
    1035219 $(x+y)^{19}$
    1135715 $(x+y)$
    1276419 $(x+y)$
    1376921 $(x+y)$
    1517619 $(x+y) $
    1678423 $(x+y)$
     | Show Table
    DownLoad: CSV

    Table 3.  Weight polynomials in references

    $p$ $k$ $d$Divisible by
    1376921 $(x+y)^{21}$
    1517619 $(x+y)^{19}$
    1678423 $(x+y)^{23}$
     | Show Table
    DownLoad: CSV

    Table 4.  Correction for $p = 127$

    $i$ $A_i$ in table $A_i$ corrected
     | Show Table
    DownLoad: CSV
  •   L. M. J. Bazzi  and  S. K. Mitter , Some randomized code constructions from group actions, IEEE Transactions on Information Theory, 52 (2006) , 3210-3219.  doi: 10.1109/TIT.2006.876244.
      L. M. J. Bazzi, Minimum Distance of Error Correcting Codes Versus Encoding Complexity, Symmetry, and Pseudorandomness, PhD thesis, MIT EECS, 2003.
      R. E. Blahut, Algebraic Codes on Lines, Planes, and Curves: An Engineering Approach, Cambridge University Press, 2008.
      I. Duursma , From weight enumerators to zeta functions, Discrete Applied Mathematics, 111 (2001) , 55-73.  doi: 10.1016/S0166-218X(00)00344-9.
      P. Gaborit , On quadratic double circulant codes over fields, Electronic Notes in Discrete Mathematics, 6 (2001) , 423-432. 
      E. N. Gilbert , A comparison of signalling alphabets, Bell System Technical Journal, 31 (1952) , 504-522.  doi: 10.1002/j.1538-7305.1952.tb01393.x.
      T. Helleseth  and  J. F. Voloch , Double circulant quadratic residue codes, IEEE Transactions on Information Theory, 50 (2004) , 2154-2155.  doi: 10.1109/TIT.2004.833371.
      D. Joyner , On quadratic residue codes and hyperelliptic curves, Discrete Mathematics and Theoretical Computer Science, 10 (2008) , 129-146. 
      D. Joyner, Selected Unsolved Problems in Coding Theory, Springer, 2011.
      M. Karlin , New binary coding results by circulants, IEEE Transactions on Information Theory, 15 (1969) , 81-92. 
      M. Larsen, The normal distribution as a limit of generalized sato-tate measures, Preprint, 1–15, URL http://mlarsen.math.indiana.edu/~larsen/unpublished.html.
      F. MacWilliams and N. Sloane, The Theory of Error-Eorrecting Codes, North Holland Publishing Co., 1977.
      Y. I. Manin , What is the maximum number of points on a curve over $F_2$, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics, 28 (1981) , 715-720. 
      E. M. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory (eds. V. S. P. Huffman and W. C.), Elsevier, 1998,177–294.
      N. J. A. Sloane, The on-line encyclopedia of integer sequences, Towards Mechanized Mathematical Assistants, 4573 (2007), 130–130, URL https://oeis.org. doi: 10.1007/978-3-540-73086-6_12.
      C. Tjhai , M. Tomlinson , M. Ambroze  and  M. Ahmed , On the Weight Distribution of the Extended Quadratic Residue Code of Prime 137, 7th International ITG Conference on Source and Channel Coding, 1 (2008) , 1-6. 
      C. Tjhai, M. Tomlinson, R. Horan, M. Ahmed and M. Ambroze, Some results on the weight distributions of the binary double-circulant codes based on primes, IEEE Singapore International Conference on Communication Systems, ICCS 2006, 2006, 1–14. doi: 10.1109/ICCS.2006.301431.
      C. J. Tjhai and M. Tomlinson, Weight distributions of quadratic residue and quadratic double circulant codes over GF(2), URL http://www.tech.plym.ac.uk/Research/fixed_and_mobile_communications/links/weightdistributions.htm.
      M. Tomlinson, C. J. Tjhai, M. A. Ambroze, M. Ahmed and M. Jibril, Error-Correction Coding and Decoding, Springer, 2017.
      R. R. Varshamov , Estimate of the number of signals in error correcting codes, Dokl. Acad. Nauk SSSR, 117 (1957) , 739-741. 
      Wikipedia, Double-precision floating-point format, 2017, URL https://en.wikipedia.org/w/index.php?title=Double-precision_floating-point_format&oldid=778810650.
  • 加载中




Article Metrics

HTML views(914) PDF downloads(478) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint