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## New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$

#### Nuh AYDİN [1] , Derek FORET [2]

Explicit construction of linear codes with best possible parameters is one of the major and challenging problems in coding theory. Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, are known to contain many codes with best known parameters. Despite the fact that these classes of codes have been extensively searched, we have been able to refine existing search algorithms to discover many new linear codes over the alphabets $\mathbb{F}_{3}$, $\mathbb{F}_{11}$, and $\mathbb{F}_{13}$ with better parameters. A total of 38 new linear codes over the three alphabets are presented.
Best known codes, Constacyclic codes, Quasi-cyclic codes, Quasi-twisted codes
• [1] R. Ackerman, N. Aydin, New quinary linear codes from quasi–twisted codes and their duals, Appl. Math. Lett. 24(4) (2011) 512–515.
• [2] N. Aydin, N. Connolly, J. Murphree, New binary linear codes from quasi–cyclic codes and an augmentation algorithm, Appl. Algebra Eng. Commun. Comput. 28(4) (2017) 339–350.
• [3] N. Aydin, N. Connolly, M. Grassl, Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi–twisted codes, Adv. Math. Commun. 11(1) (2017) 245–258.
• [4] N. Aydin, J. Lambrinos, O. VandenBerg, On equivalence of cyclic codes, generalization of a quasi– twisted search algorithm, and new linear codes, in submission.
• [5] N. Aydin, J. M. Murphree, New linear codes from constacyclic codes, J. Frankl. Inst. 351(3) (2014) 1691–1699.
• [6] N. Aydin, I. Siap, D. K. Ray-Chaudhuri, The structure of 1–generator quasi–twisted codes and new linear codes, Des. Codes Cryptogr. 24(3) (2001) 313–326.
• [7] N. Aydin, I. Siap, New quasi–cyclic codes over $\mathbb{F}_5$, Appl. Math. Lett. 15(7) (2002) 833–836.
• [8] W. Bosma, J. J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbol. Comput. 24(3–4) (1997) 235–265.
• [9] 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 Appl. 2(1) (2015) 1–16.
• [10] E. Z. Chen, N. Aydin, New quasi-twisted codes over $F_{11}$–minimum distance bounds and a new database, J. Inform. Optim. Sci. 36(1–2) (2015) 129–157.
• [11] E. Z. Chen, Six new binary quasi–cyclic codes, IEEE Trans. Inform. Theory 40(5) (1994) 1666–1667.
• [12] R. Daskalov, P. Hristov, New binary one–generator quasi–cyclic codes, IEEE Trans. Inform. Theory 49(11) (2003) 3001–3005.
• [13] R. Daskalov, P. Hristov, E. Metodieva, New minimum distance bounds for linear codes over GF(5), Discrete Math. 275(1-3) (2004) 97–110.
• [14] M. Grassl, Code Tables: Bounds on the minimum distance of linear codes and quantum codes, online, http://www.codetables.de/
• [15] T. A. Gulliver, V. K. Bhargava, New good rate $(m-1)/pm$ ternary and quaternary quasi–cyclic codes, Des. Codes Cryptogr. 7(3) (196) 223–233.
• [16] I. Siap, N. Aydin, D. K. Ray–Chaudhuri, New ternary quasi–cyclic codes with better minimum distances, IEEE Trans. Inform. Theory 46(4) (2000) 1554–1558.
• [17] A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory 43(6) (1997) 1757–1766.
Primary Language en Engineering Articles Orcid: 0000-0002-5618-2427Author: Nuh AYDİN (Primary Author) Author: Derek FORET Publication Date : January 19, 2019
 Bibtex @research article { jacodesmath508968, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2019}, volume = {6}, pages = {13 - 20}, doi = {10.13069/jacodesmath.508968}, title = {New Linear Codes over \$GF(3)\$, \$GF(11)\$, and \$GF(13)\$}, key = {cite}, author = {Aydi̇n, Nuh and Foret, Derek} } APA Aydi̇n, N , Foret, D . (2019). New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$ . Journal of Algebra Combinatorics Discrete Structures and Applications , 6 (1) , 13-20 . DOI: 10.13069/jacodesmath.508968 MLA Aydi̇n, N , Foret, D . "New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$" . Journal of Algebra Combinatorics Discrete Structures and Applications 6 (2019 ): 13-20 Chicago Aydi̇n, N , Foret, D . "New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$". Journal of Algebra Combinatorics Discrete Structures and Applications 6 (2019 ): 13-20 RIS TY - JOUR T1 - New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$ AU - Nuh Aydi̇n , Derek Foret Y1 - 2019 PY - 2019 N1 - doi: 10.13069/jacodesmath.508968 DO - 10.13069/jacodesmath.508968 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 13 EP - 20 VL - 6 IS - 1 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.508968 UR - https://doi.org/10.13069/jacodesmath.508968 Y2 - 2018 ER - EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$ %A Nuh Aydi̇n , Derek Foret %T New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$ %D 2019 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 6 %N 1 %R doi: 10.13069/jacodesmath.508968 %U 10.13069/jacodesmath.508968 ISNAD Aydi̇n, Nuh , Foret, Derek . "New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$". Journal of Algebra Combinatorics Discrete Structures and Applications 6 / 1 (January 2019): 13-20 . https://doi.org/10.13069/jacodesmath.508968 AMA Aydi̇n N , Foret D . New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2019; 6(1): 13-20. Vancouver Aydi̇n N , Foret D . New Linear Codes over $GF(3)$, $GF(11)$, and $GF(13)$. Journal of Algebra Combinatorics Discrete Structures and Applications. 2019; 6(1): 13-20.

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Authors of the Article
[2]