    May  2018, 12(2): 363-385. doi: 10.3934/amc.2018023

## The weight distribution of quasi-quadratic residue codes

 1 Department of Mathematics, Department of Electric and Computer Engineering, University of Wisconsin-Madison, WI 53706, United States 2 Department of Mathematics, University of Wisconsin-Madison, WI 53706, United States

Received  May 2017 Published  March 2018

Fund Project: The first author is supported by Simons Foundation grant MSN179747.

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.

Citation: Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023
##### References:
  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.  Google Scholar  L. M. J. Bazzi, Minimum Distance of Error Correcting Codes Versus Encoding Complexity, Symmetry, and Pseudorandomness, PhD thesis, MIT EECS, 2003. Google Scholar  R. E. Blahut, Algebraic Codes on Lines, Planes, and Curves: An Engineering Approach, Cambridge University Press, 2008. Google Scholar  I. Duursma, From weight enumerators to zeta functions, Discrete Applied Mathematics, 111 (2001), 55-73.  doi: 10.1016/S0166-218X(00)00344-9.  Google Scholar  P. Gaborit, On quadratic double circulant codes over fields, Electronic Notes in Discrete Mathematics, 6 (2001), 423-432. Google Scholar  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. Google Scholar  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.  Google Scholar  D. Joyner, On quadratic residue codes and hyperelliptic curves, Discrete Mathematics and Theoretical Computer Science, 10 (2008), 129-146. Google Scholar  D. Joyner, Selected Unsolved Problems in Coding Theory, Springer, 2011. Google Scholar  M. Karlin, New binary coding results by circulants, IEEE Transactions on Information Theory, 15 (1969), 81-92. Google Scholar  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. Google Scholar  F. MacWilliams and N. Sloane, The Theory of Error-Eorrecting Codes, North Holland Publishing Co., 1977. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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.   Google Scholar  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. Google Scholar  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. Google Scholar  M. Tomlinson, C. J. Tjhai, M. A. Ambroze, M. Ahmed and M. Jibril, Error-Correction Coding and Decoding, Springer, 2017. Google Scholar  R. R. Varshamov, Estimate of the number of signals in error correcting codes, Dokl. Acad. Nauk SSSR, 117 (1957), 739-741. Google Scholar  Wikipedia, Double-precision floating-point format, 2017, URL https://en.wikipedia.org/w/index.php?title=Double-precision_floating-point_format&oldid=778810650. Google Scholar

show all references

##### References:
  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.  Google Scholar  L. M. J. Bazzi, Minimum Distance of Error Correcting Codes Versus Encoding Complexity, Symmetry, and Pseudorandomness, PhD thesis, MIT EECS, 2003. Google Scholar  R. E. Blahut, Algebraic Codes on Lines, Planes, and Curves: An Engineering Approach, Cambridge University Press, 2008. Google Scholar  I. Duursma, From weight enumerators to zeta functions, Discrete Applied Mathematics, 111 (2001), 55-73.  doi: 10.1016/S0166-218X(00)00344-9.  Google Scholar  P. Gaborit, On quadratic double circulant codes over fields, Electronic Notes in Discrete Mathematics, 6 (2001), 423-432. Google Scholar  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. Google Scholar  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.  Google Scholar  D. Joyner, On quadratic residue codes and hyperelliptic curves, Discrete Mathematics and Theoretical Computer Science, 10 (2008), 129-146. Google Scholar  D. Joyner, Selected Unsolved Problems in Coding Theory, Springer, 2011. Google Scholar  M. Karlin, New binary coding results by circulants, IEEE Transactions on Information Theory, 15 (1969), 81-92. Google Scholar  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. Google Scholar  F. MacWilliams and N. Sloane, The Theory of Error-Eorrecting Codes, North Holland Publishing Co., 1977. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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.   Google Scholar  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. Google Scholar  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. Google Scholar  M. Tomlinson, C. J. Tjhai, M. A. Ambroze, M. Ahmed and M. Jibril, Error-Correction Coding and Decoding, Springer, 2017. Google Scholar  R. R. Varshamov, Estimate of the number of signals in error correcting codes, Dokl. Acad. Nauk SSSR, 117 (1957), 739-741. Google Scholar  Wikipedia, Double-precision floating-point format, 2017, URL https://en.wikipedia.org/w/index.php?title=Double-precision_floating-point_format&oldid=778810650. Google Scholar
Computational Results
 $p$ $d$ $\delta$ Divisible by 3 2 0.33 $(x^2 + y^2)^3$ 11 6 0.27 $(x^2 + y^2)^7$ 19 8 0.21 $(x^2 + y^2)^7$ 43 14 0.16 $(x^2 + y^2)^{15}$ 59 18 0.15 $(x^2 + y^2)^{19}$ 67 22 0.16 $(x^2 + y^2)^{23}$
 $p$ $d$ $\delta$ Divisible by 3 2 0.33 $(x^2 + y^2)^3$ 11 6 0.27 $(x^2 + y^2)^7$ 19 8 0.21 $(x^2 + y^2)^7$ 43 14 0.16 $(x^2 + y^2)^{15}$ 59 18 0.15 $(x^2 + y^2)^{19}$ 67 22 0.16 $(x^2 + y^2)^{23}$
Weight polynomials posted on 
 $p$ $k$ $d$ Divisible by 89 45 17 $(x+y)^{17}$ 97 49 15 $(x+y)^{15}$ 103 52 19 $(x+y)^{19}$ 113 57 15 $(x+y)$ 127 64 19 $(x+y)$ 137 69 21 $(x+y)$ 151 76 19 $(x+y)$ 167 84 23 $(x+y)$
 $p$ $k$ $d$ Divisible by 89 45 17 $(x+y)^{17}$ 97 49 15 $(x+y)^{15}$ 103 52 19 $(x+y)^{19}$ 113 57 15 $(x+y)$ 127 64 19 $(x+y)$ 137 69 21 $(x+y)$ 151 76 19 $(x+y)$ 167 84 23 $(x+y)$
Weight polynomials in references
 $p$ $k$ $d$ Divisible by 137 69 21 $(x+y)^{21}$ 151 76 19 $(x+y)^{19}$ 167 84 23 $(x+y)^{23}$
 $p$ $k$ $d$ Divisible by 137 69 21 $(x+y)^{21}$ 151 76 19 $(x+y)^{19}$ 167 84 23 $(x+y)^{23}$
Correction for $p = 127$
 $i$ $A_i$ in table $A_i$ corrected 51 223367511592873280 223367511592873284 52 326460209251122496 326460209251122492 55 840260234424082176 840260234424082220 56 1080334587116677120 1080334587116677140 59 1899366974583683328 1899366974583683220 60 2152615904528174336 2152615904528174316 63 2596788489999036416 2596788489999036307
 $i$ $A_i$ in table $A_i$ corrected 51 223367511592873280 223367511592873284 52 326460209251122496 326460209251122492 55 840260234424082176 840260234424082220 56 1080334587116677120 1080334587116677140 59 1899366974583683328 1899366974583683220 60 2152615904528174336 2152615904528174316 63 2596788489999036416 2596788489999036307
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