[1] N. Aydin, I. Siap, D. K. Ray–Chaudhuri, The structure of 1–generator quasi–twisted codes and new
linear codes, Des. Codes Cryptogr. 24 (2001) 313–326.
[2] N. Aydin, I. Siap, New quasi–cyclic codes over F$_{5}$, Appl. Math. Lett. 15 (2002) 833–836.
[3] N. Aydin, J. Murphree, New linear codes from constacyclic codes, J. Franklin Inst. 351(3) (2014)
1691–1699.
[4] N. Aydin, N. Connolly, M. Grassl, Some results on the structure of constacyclic codes and new linear
codes over GF(7) from QT codes, Adv. Math. Commun. 11(1) (2017) 245–258.
[5] N. Aydin, N. Connolly, J. Murphree, New binary linear codes from quasi–cyclic codes and an augmentation
algorithm, Appl. Algebra Engrg. Comm. Comput. 28(4) (2017) 339–350.
[6] N. Aydin,J. Lambrinos, O. VandenBerg, On equivalence of cyclic codes, generalization of a quasi–
twisted search algorithm, and new linear codes, Des. Codes Cryptogr. 87 (2019) 2199–2212.
[7] N. Aydin, D. Foret, New linear codes over GF(3), GF(11) and GF(13), J. Algebra Comb. Discrete
Struct. Appl. 6(1) (2019) 13–20.
[8] S. Ball, Table of bounds on three dimensional linear codes or $(n, r)$ Arcs in PG(2, q), available at
https://web.mat.upc.edu/simeon.michael.ball/codebounds.html
[9] E. Z. Chen, Database of Quasi–twisted codes, available at
http://www.tec.hkr.se/chen/research/codes/
[10] E. Z. Chen, A new iterative computer search algorithm for good quasi–twisted codes,Des. Codes
Cryptogr. 76(2) (2015) 307–323.
[11] E. Z. Chen, Some new binary linear codes with improved minimum distances, J. Algebra Comb.
Discrete Struct. Appl. 5(2) (2018) 65–70.
[12] E. Z. Chen, N. Aydin, New quasi–twisted codes over F$_{11}$ minimum distance bounds and a new
database, J. Inf. Optim. Sci. 36(1–2) (2015) 129–157.
[13] E. Z. Chen, N. Aydin, A database of linear codes over F$_{13}$ with minimum distance bounds and new
quasi–twisted codes from a heuristic search algorithm, J. Algebra Comb. Discrete Struct. Appl. 2(1)
(2015) 1–16.
[14] R. Daskalov, E. Metodieva, The nonexistence of some 5–dimensional quaternary linear codes, IEEE
Trans. Inform. Theory 41(2) (1995) 581–583.
[15] R. N. Daskalov, T. A. Gulliver, New good quasi–cyclic ternary and quaternary linear codes, IEEE
Trans. Inform. Theory 43 (1997) 1647–1650.
[16] R. Daskalov, T. A. Gulliver, Bounds on Minimum Distance for Linear Codes over GF(5), Appl.
Algebra Engrg. Comm. Comput. 9(6) (1999) 547–558.
[17] R. Daskalov, P. Hristov, New one–enerator quasi–cyclic codes over GF(7), Problemi Peredachi Informatsii
38(1) (2002) 59–63. English translation: Probl. Inf. Transm. 38(1) (2002) 50–54.
[18] R. Daskalov, P. Hristov, Some new quasi–twisted ternary linear codes, J. Algebra Comb. Discrete
Struct. Appl. 2(3) (2016) 211–216.
[19] R. Daskalov, P. Hristov, Some new ternary linear codes, J. Algebra Comb. Discrete Struct. Appl.
4(3) (2017) 227–234.
[20] R. Daskalov, P. Hristov, E. Metodieva, New minimum distance bounds for linear codes over GF(5),
Discrete Math. 275(1–3) (2004) 97–110.
[21] R. Daskalov, E. Metodieva, The nonexistence of ternary [105,6,68] and [230,6,152] codes, Discrete
Math. 286(3) (2004) 225–232.
[22] H. Fredricksen, J. Maiorana, Necklaces of beads in k colors and k–ary de Bruijn sequences, Discrete
Math. 23 (1978) 207–210.
[23] H. Fredricksen, I. J. Kessler, An algorithm for generating necklaces of beads in two colors, Discrete
Math. 61 (1986) 181–188.
[24] P. P. Greenough, R. Hill, Optimal ternary quasi–cyclic codes, Des. Codes Crypt. 2 (1992) 81–91.
[25] T. A. Gulliver, Quasi–twisted codes over F$_{11}$ , Ars Combinatoria 99 (2011) 3–17.
[26] T. A. Gulliver, Quasi–cyclic codes over F$_{13}$ , In Combinatorial Algorithms, Lecture Notes in Computer
Science 7056 (2011) 236–246.
[27] T. A. Gulliver, P. R. J. Ostergard, Improved bounds for ternary linear codes of dimension 7, IEEE
Trans. Inform. Theory 43 (1997) 1377–1388.
[28] M. Grassl, Bounds on the minimum distances of linear codes, available at http://www.codetables.de
[29] F. Ruskey, C. Savage, T. Wang, Generating necklaces, Journal of Algorithms 13 (1992) 414–430.
[30] A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform.
Theory 43 (1997) 1757–1766.
[31] V. Venkaiah, T. A.Gulliver, Quasi–cyclic codes over F13 and enumeration of definining polynomials,
Journal Discrete Algorithms 16 (2012) 249–257.
[32] T. A. Gulliver, V. Venkaiah, Construction of quasi–twisted codes and enumeration of definining
polynomials, J. Algebra Comb. Discrete Struct. Appl. 7(1) (2020) 3–20.
Generating generalized necklaces and new quasi-cyclic codes
In many cases there is a need of exhaustive lists of combinatorial objects of a given type. We consider generation of all inequivalent
polynomials from which defining polynomials for constructing quasi-cyclic (QC) codes are to be chosen. Using these defining polynomials we construct 34 new good QC codes over GF(11) and 36 such codes over GF(13). In many cases there is a need of exhaustive lists of combinatorial objects of a given type. We consider generation of all inequivalent polynomials from which defining polynomials for constructing quasi-cyclic (QC) codes are to be chosen. Using these defining polynomials we construct 34 new good QC codes over GF(11) and 36 such codes over GF(13).
[1] N. Aydin, I. Siap, D. K. Ray–Chaudhuri, The structure of 1–generator quasi–twisted codes and new
linear codes, Des. Codes Cryptogr. 24 (2001) 313–326.
[2] N. Aydin, I. Siap, New quasi–cyclic codes over F$_{5}$, Appl. Math. Lett. 15 (2002) 833–836.
[3] N. Aydin, J. Murphree, New linear codes from constacyclic codes, J. Franklin Inst. 351(3) (2014)
1691–1699.
[4] N. Aydin, N. Connolly, M. Grassl, Some results on the structure of constacyclic codes and new linear
codes over GF(7) from QT codes, Adv. Math. Commun. 11(1) (2017) 245–258.
[5] N. Aydin, N. Connolly, J. Murphree, New binary linear codes from quasi–cyclic codes and an augmentation
algorithm, Appl. Algebra Engrg. Comm. Comput. 28(4) (2017) 339–350.
[6] N. Aydin,J. Lambrinos, O. VandenBerg, On equivalence of cyclic codes, generalization of a quasi–
twisted search algorithm, and new linear codes, Des. Codes Cryptogr. 87 (2019) 2199–2212.
[7] N. Aydin, D. Foret, New linear codes over GF(3), GF(11) and GF(13), J. Algebra Comb. Discrete
Struct. Appl. 6(1) (2019) 13–20.
[8] S. Ball, Table of bounds on three dimensional linear codes or $(n, r)$ Arcs in PG(2, q), available at
https://web.mat.upc.edu/simeon.michael.ball/codebounds.html
[9] E. Z. Chen, Database of Quasi–twisted codes, available at
http://www.tec.hkr.se/chen/research/codes/
[10] E. Z. Chen, A new iterative computer search algorithm for good quasi–twisted codes,Des. Codes
Cryptogr. 76(2) (2015) 307–323.
[11] E. Z. Chen, Some new binary linear codes with improved minimum distances, J. Algebra Comb.
Discrete Struct. Appl. 5(2) (2018) 65–70.
[12] E. Z. Chen, N. Aydin, New quasi–twisted codes over F$_{11}$ minimum distance bounds and a new
database, J. Inf. Optim. Sci. 36(1–2) (2015) 129–157.
[13] E. Z. Chen, N. Aydin, A database of linear codes over F$_{13}$ with minimum distance bounds and new
quasi–twisted codes from a heuristic search algorithm, J. Algebra Comb. Discrete Struct. Appl. 2(1)
(2015) 1–16.
[14] R. Daskalov, E. Metodieva, The nonexistence of some 5–dimensional quaternary linear codes, IEEE
Trans. Inform. Theory 41(2) (1995) 581–583.
[15] R. N. Daskalov, T. A. Gulliver, New good quasi–cyclic ternary and quaternary linear codes, IEEE
Trans. Inform. Theory 43 (1997) 1647–1650.
[16] R. Daskalov, T. A. Gulliver, Bounds on Minimum Distance for Linear Codes over GF(5), Appl.
Algebra Engrg. Comm. Comput. 9(6) (1999) 547–558.
[17] R. Daskalov, P. Hristov, New one–enerator quasi–cyclic codes over GF(7), Problemi Peredachi Informatsii
38(1) (2002) 59–63. English translation: Probl. Inf. Transm. 38(1) (2002) 50–54.
[18] R. Daskalov, P. Hristov, Some new quasi–twisted ternary linear codes, J. Algebra Comb. Discrete
Struct. Appl. 2(3) (2016) 211–216.
[19] R. Daskalov, P. Hristov, Some new ternary linear codes, J. Algebra Comb. Discrete Struct. Appl.
4(3) (2017) 227–234.
[20] R. Daskalov, P. Hristov, E. Metodieva, New minimum distance bounds for linear codes over GF(5),
Discrete Math. 275(1–3) (2004) 97–110.
[21] R. Daskalov, E. Metodieva, The nonexistence of ternary [105,6,68] and [230,6,152] codes, Discrete
Math. 286(3) (2004) 225–232.
[22] H. Fredricksen, J. Maiorana, Necklaces of beads in k colors and k–ary de Bruijn sequences, Discrete
Math. 23 (1978) 207–210.
[23] H. Fredricksen, I. J. Kessler, An algorithm for generating necklaces of beads in two colors, Discrete
Math. 61 (1986) 181–188.
[24] P. P. Greenough, R. Hill, Optimal ternary quasi–cyclic codes, Des. Codes Crypt. 2 (1992) 81–91.
[25] T. A. Gulliver, Quasi–twisted codes over F$_{11}$ , Ars Combinatoria 99 (2011) 3–17.
[26] T. A. Gulliver, Quasi–cyclic codes over F$_{13}$ , In Combinatorial Algorithms, Lecture Notes in Computer
Science 7056 (2011) 236–246.
[27] T. A. Gulliver, P. R. J. Ostergard, Improved bounds for ternary linear codes of dimension 7, IEEE
Trans. Inform. Theory 43 (1997) 1377–1388.
[28] M. Grassl, Bounds on the minimum distances of linear codes, available at http://www.codetables.de
[29] F. Ruskey, C. Savage, T. Wang, Generating necklaces, Journal of Algorithms 13 (1992) 414–430.
[30] A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform.
Theory 43 (1997) 1757–1766.
[31] V. Venkaiah, T. A.Gulliver, Quasi–cyclic codes over F13 and enumeration of definining polynomials,
Journal Discrete Algorithms 16 (2012) 249–257.
[32] T. A. Gulliver, V. Venkaiah, Construction of quasi–twisted codes and enumeration of definining
polynomials, J. Algebra Comb. Discrete Struct. Appl. 7(1) (2020) 3–20.
Daskalov, R., & Metodıeva, E. (2020). Generating generalized necklaces and new quasi-cyclic codes. Journal of Algebra Combinatorics Discrete Structures and Applications, 7(3), 237-245. https://doi.org/10.13069/jacodesmath.784999
AMA
Daskalov R, Metodıeva E. Generating generalized necklaces and new quasi-cyclic codes. Journal of Algebra Combinatorics Discrete Structures and Applications. September 2020;7(3):237-245. doi:10.13069/jacodesmath.784999
Chicago
Daskalov, Rumen, and Elena Metodıeva. “Generating Generalized Necklaces and New Quasi-Cyclic Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 7, no. 3 (September 2020): 237-45. https://doi.org/10.13069/jacodesmath.784999.
EndNote
Daskalov R, Metodıeva E (September 1, 2020) Generating generalized necklaces and new quasi-cyclic codes. Journal of Algebra Combinatorics Discrete Structures and Applications 7 3 237–245.
IEEE
R. Daskalov and E. Metodıeva, “Generating generalized necklaces and new quasi-cyclic codes”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 3, pp. 237–245, 2020, doi: 10.13069/jacodesmath.784999.
ISNAD
Daskalov, Rumen - Metodıeva, Elena. “Generating Generalized Necklaces and New Quasi-Cyclic Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 7/3 (September 2020), 237-245. https://doi.org/10.13069/jacodesmath.784999.
JAMA
Daskalov R, Metodıeva E. Generating generalized necklaces and new quasi-cyclic codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7:237–245.
MLA
Daskalov, Rumen and Elena Metodıeva. “Generating Generalized Necklaces and New Quasi-Cyclic Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 3, 2020, pp. 237-45, doi:10.13069/jacodesmath.784999.
Vancouver
Daskalov R, Metodıeva E. Generating generalized necklaces and new quasi-cyclic codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7(3):237-45.