# American Institute of Mathematical Sciences

November  2018, 12(4): 659-679. doi: 10.3934/amc.2018039

## On the dual codes of skew constacyclic codes

 1 Universidad de Concepción, Escuela de Educación, Departamento de Ciencias Básicas, Los Ángeles, Chile 2 Universidad de Concepción, Facultad de Ciencias Físicas y Matemáticas, Departamento de Matemática, Concepción, Chile

* Corresponding author: Andrea L. Tironi

Received  March 2017 Revised  April 2018 Published  September 2018

Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $\theta : \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with skew constacyclic codes, that is, linear codes of $\mathbb{F}_q^n$ which are invariant under the action of a semi-linear map $\phi _{\alpha,\theta }:\mathbb{F}_q^n\to\mathbb{F}_q^n$, defined by $\phi _{\alpha,\theta }(a_0,...,a_{n-2}, a_{n-1}): = (\alpha \theta (a_{n-1}),\theta (a_0),...,\theta (a_{n-2}))$ for some $\alpha \in \mathbb{F}_q\setminus\{0\}$ and $n≥2$. In particular, we study some algebraic and geometric properties of their dual codes and we give some consequences and research results on $1$-generator skew quasi-twisted codes and on MDS skew constacyclic codes.

Citation: Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039
##### References:
 [1] T. Abualrub, A. Ghrayeb, N. Aydin and I. Siap, On the construction of skew quasi-cyclic codes, IEEE Trans. Inform. Theory, 56 (2010), 2081-2090. doi: 10.1109/TIT.2010.2044062. Google Scholar [2] M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101. doi: 10.1007/s10623-011-9494-0. Google Scholar [3] A. Blokhuis, A. A. Bruen and J. A. Thas, Arcs in $PG(n,q)$, MDS-codes and three fundamental problems of B. Segre - some extensions, Geom. Dedic., 35 (1990), 1-11. doi: 10.1007/BF00147336. Google Scholar [4] A. Blokhuis, A. A. Bruen and J. A. Thas, On MDS-codes, arcs in $PG(n,q)$ with $q$ even, and a solution of three fundamental problems of B. Segre, Invent. Math., 92 (1988), 441-459. doi: 10.1007/BF01393742. Google Scholar [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125. Google Scholar [6] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z. Google Scholar [7] D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292. doi: 10.3934/amc.2008.2.273. Google Scholar [8] D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, Cryptography and coding, Lecture Notes in Comput. Sci., 5921, Springer, Berlin, 2009, 38–55. doi: 10.1007/978-3-642-10868-6_3. Google Scholar [9] D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, Cryptography and Coding, Lecture Notes in Comput. Sci., 7089, Springer, Heidelberg, 2011,230–243. doi: 10.1007/978-3-642-25516-8_14. Google Scholar [10] D. Boucher and F. Ulmer, Self-dual skew codes and factorization of skew polynomials, J. Symbolic Comput., 60 (2014), 47-61. doi: 10.1016/j.jsc.2013.10.003. Google Scholar [11] M'Hammed Boulagouaz and A. Leroy, (σ, δ)-codes, Adv. Math. Commun., 7 (2013), 463-474. doi: 10.3934/amc.2013.7.463. Google Scholar [12] A. Cherchem and A. Leroy, Exponents of skew polynomials, Finite Fields and Their Appl., 37 (2016), 1-13. doi: 10.1016/j.ffa.2015.08.004. Google Scholar [13] N. Fogarty and H. Gluesing-Luerssen, A circulant approach to skew-constacyclic codes, Finite Fields Appl., 35 (2015), 92-114. doi: 10.1016/j.ffa.2015.03.008. Google Scholar [14] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, Claredon Press - Oxford, 1979. Google Scholar [15] S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63. doi: 10.3934/amc.2012.6.39. Google Scholar [16] H. Kaneta and T. Maruta, An elementary proof and extension of Thas' theorem on k-arcs, Mat. Proc. Camb. Philos. Soc., 105 (1989), 459-462. doi: 10.1017/S0305004100077823. Google Scholar [17] A. Leroy, Noncommutative polynomial maps, J. Algebra Appl., 11 (2012), 1250076, 16 pp. doi: 10.1142/S0219498812500764. Google Scholar [18] T. Maruta, A geometric approach to semi-cyclic codes, Advances in Finite Geometries and Designs (Chelwood Gate, 1990), Oxford Sci. Publ., Oxford Univ. Press, New York, (1991), 311–318. Google Scholar [19] T. Maruta, On the existence of cyclic and pseudo-cyclic MDS codes, Europ. J. Combinatorics, 19 (1998), 159-174. doi: 10.1006/S0195-6698(97)90000-7. Google Scholar [20] T. Maruta, M. Shinohara and M. Takenaka, Constructing linear codes from some orbits of projectivities, Discrete Math., 308 (2008), 832-841. doi: 10.1016/j.disc.2007.07.045. Google Scholar [21] L. Storme and J. A. Thas, M.D.S. codes and arcs in $PG(n,q)$ with $q$ even: An improvement of the bounds of Bruen, Thas, and Blokhuis, J. Comb. Theory, Ser. A, 62 (1993), 139-154. doi: 10.1016/0097-3165(93)90076-K. Google Scholar [22] L. F. Tapia Cuitiño and A. L. Tironi, Dual codes of product semi-linear codes, Linear Algebra Appl., 457 (2014), 114-153. doi: 10.1016/j.laa.2014.05.011. Google Scholar [23] L. F. Tapia Cuitiño and A. L. Tironi, Some properties of skew codes over finite fields, Des. Codes Cryptogr., 85 (2017), 359-380. doi: 10.1007/s10623-016-0311-7. Google Scholar [24] J. A. Thas, Normal rational curves and $(q+2)$-arcs in a Galois space $S_{q-2,q}(q=2^h)$, Atti Accad. Naz. Lincei Rend., 47 (1969), 249-252. Google Scholar

show all references

##### References:
 [1] T. Abualrub, A. Ghrayeb, N. Aydin and I. Siap, On the construction of skew quasi-cyclic codes, IEEE Trans. Inform. Theory, 56 (2010), 2081-2090. doi: 10.1109/TIT.2010.2044062. Google Scholar [2] M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101. doi: 10.1007/s10623-011-9494-0. Google Scholar [3] A. Blokhuis, A. A. Bruen and J. A. Thas, Arcs in $PG(n,q)$, MDS-codes and three fundamental problems of B. Segre - some extensions, Geom. Dedic., 35 (1990), 1-11. doi: 10.1007/BF00147336. Google Scholar [4] A. Blokhuis, A. A. Bruen and J. A. Thas, On MDS-codes, arcs in $PG(n,q)$ with $q$ even, and a solution of three fundamental problems of B. Segre, Invent. Math., 92 (1988), 441-459. doi: 10.1007/BF01393742. Google Scholar [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125. Google Scholar [6] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z. Google Scholar [7] D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292. doi: 10.3934/amc.2008.2.273. Google Scholar [8] D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, Cryptography and coding, Lecture Notes in Comput. Sci., 5921, Springer, Berlin, 2009, 38–55. doi: 10.1007/978-3-642-10868-6_3. Google Scholar [9] D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, Cryptography and Coding, Lecture Notes in Comput. Sci., 7089, Springer, Heidelberg, 2011,230–243. doi: 10.1007/978-3-642-25516-8_14. Google Scholar [10] D. Boucher and F. Ulmer, Self-dual skew codes and factorization of skew polynomials, J. Symbolic Comput., 60 (2014), 47-61. doi: 10.1016/j.jsc.2013.10.003. Google Scholar [11] M'Hammed Boulagouaz and A. Leroy, (σ, δ)-codes, Adv. Math. Commun., 7 (2013), 463-474. doi: 10.3934/amc.2013.7.463. Google Scholar [12] A. Cherchem and A. Leroy, Exponents of skew polynomials, Finite Fields and Their Appl., 37 (2016), 1-13. doi: 10.1016/j.ffa.2015.08.004. Google Scholar [13] N. Fogarty and H. Gluesing-Luerssen, A circulant approach to skew-constacyclic codes, Finite Fields Appl., 35 (2015), 92-114. doi: 10.1016/j.ffa.2015.03.008. Google Scholar [14] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, Claredon Press - Oxford, 1979. Google Scholar [15] S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63. doi: 10.3934/amc.2012.6.39. Google Scholar [16] H. Kaneta and T. Maruta, An elementary proof and extension of Thas' theorem on k-arcs, Mat. Proc. Camb. Philos. Soc., 105 (1989), 459-462. doi: 10.1017/S0305004100077823. Google Scholar [17] A. Leroy, Noncommutative polynomial maps, J. Algebra Appl., 11 (2012), 1250076, 16 pp. doi: 10.1142/S0219498812500764. Google Scholar [18] T. Maruta, A geometric approach to semi-cyclic codes, Advances in Finite Geometries and Designs (Chelwood Gate, 1990), Oxford Sci. Publ., Oxford Univ. Press, New York, (1991), 311–318. Google Scholar [19] T. Maruta, On the existence of cyclic and pseudo-cyclic MDS codes, Europ. J. Combinatorics, 19 (1998), 159-174. doi: 10.1006/S0195-6698(97)90000-7. Google Scholar [20] T. Maruta, M. Shinohara and M. Takenaka, Constructing linear codes from some orbits of projectivities, Discrete Math., 308 (2008), 832-841. doi: 10.1016/j.disc.2007.07.045. Google Scholar [21] L. Storme and J. A. Thas, M.D.S. codes and arcs in $PG(n,q)$ with $q$ even: An improvement of the bounds of Bruen, Thas, and Blokhuis, J. Comb. Theory, Ser. A, 62 (1993), 139-154. doi: 10.1016/0097-3165(93)90076-K. Google Scholar [22] L. F. Tapia Cuitiño and A. L. Tironi, Dual codes of product semi-linear codes, Linear Algebra Appl., 457 (2014), 114-153. doi: 10.1016/j.laa.2014.05.011. Google Scholar [23] L. F. Tapia Cuitiño and A. L. Tironi, Some properties of skew codes over finite fields, Des. Codes Cryptogr., 85 (2017), 359-380. doi: 10.1007/s10623-016-0311-7. Google Scholar [24] J. A. Thas, Normal rational curves and $(q+2)$-arcs in a Galois space $S_{q-2,q}(q=2^h)$, Atti Accad. Naz. Lincei Rend., 47 (1969), 249-252. Google Scholar
New constructions of some linear codes with BKLC parameters
 $[n,k,d]_q$ Generator Matrix $N$ $\alpha$ $p^t$ $[25,4,17]_4$ $[1a^21a^2]^5+[a^2a^2a^21]^{10}+[aa^210]^{10}$ 5 $a$ 2 $[35,4,24]_4$ $[1a1a]^5+[0a^2a1]^{10}+[10a^21]^{10}+[1a^201]^{10}$ 5 $a^2$ 2 $[40,4,28]_4$ $[1a^21a^2]^5+[0a^2a1]^{10}+[a^2a^201]^{10}+[1a^201]^{10}+[1010]^5$ 5 $a$ 2 $[45,4,32]_4$ $[1a1a]^5+[a100]^{10}+[1a^211]^{10}+[0a01]^{10}+[a^2a^2a1]^{10}$ 5 $a^2$ 2 $[50,4,36]_4$ $[1a1a]^5+[1aa^21]^{10}+[1101]^{10}+[aaa^21]^{10}+[a^2a^2a1]^{10}+[a^2100]^5$ 5 $a^2$ 2 $[60,4,44]_4$ $[1a^21a^2]^5+[1110]^{10}+[0aa^21]^{10}+[00a^21]^{10}+[a^2111]^{10} +[a^2a^211]^{10}+[a0a1]^5$ 5 $a$ 2 $[65,4,48]_4$ $[1a1a]^5+[aa01]^{10}+[1011]^{10}+[a1a^21]^{10} +[1a^210]^{10}+[0a11]^{10}+[a^2100]^5+[a^20a^21]^{5}$ 5 $a^2$ 2 $[85,4,64]_4$ $[1a1a]^5+[a^2a11]^{10}+[0011]^{10}+[a^2a^201]^{10}+[1a^201]^{10}+[0a^2a^21]^{10}+[a^21a1]^{10}+[1a^2a^21]^{10}+[1010]^{5}+[0a10]^5$ 5 $a^2$ 2 $[12, 5, 6]_4$ $[a^21a^21a^2]^{6}+[1a^2a^2aa^2]^{6}$ 6 1 2 $[18, 5, 10]_4$ $[a^21a^21a^2]^{6}+[01a^201]^{6}+[a^20a^2aa]^{6}$ 6 1 2 $[20, 5, 12]_4$ $[aa110]^{10}+[a^20a^2aa]^{10}$ 10 1 2 $[30, 5,20]_4$ $[11a^2a1]^{10}+[a^2aaa^21]^{10}+[01a^201]^{10}$ 10 1 2 $[32, 5, 21]_4$ $[a^2aa^2a^20]^{8}+[aa1a^20]^{8}+[0a^20a^2a]^{8}+[1001a^2]^{8}$ 8 1 2 $[36, 5, 24]_4$ $[a^2aa^20a^2]^{12}+[a^2aaa^21]^{12}+[01a^201]^{12}$ 12 1 2 $[48, 5, 33]_4$ $[1101a^2]^{12}+[a^20a^2aa]^{12}+[0a^20a^2a]^{12}+[a^2a^2a^21a]^{12}$ 12 1 2 $[120, 5, 88]_4$ $[a^21a1a^2]^{30}+[a^2aaa^21]^{30}+[a^20a^2aa]^{30}+[0a^200a]^{30}$ 30 1 2 $[248, 5,184]_4$ $[aa^2a^2aa]^{62}+[a^2aaa^21]^{62}+[aa1a^20]^{62}+[0a^200a]^{62}$ 62 1 2 $[21,6,12]_4$ $[1a^21a^21a^2]^7+[aa0a11]^{14}$ $7$ $a$ 2 $[49,6,32]_4$ $[1a^21a^21a^2]^7+[aa^2a^2101]^{14}+[a^21aaa1]^{14}+[aa^21a^210]^{14}$ 7 $a$ 2 $[34,3,28]_8$ $[1w^6w^4]^4+[w^6w1]^{12}+[w^510]^{12}+[111]^6$ 4 $w$ 2 $[50,4,41]_8$ $[w^51w^4w^5]^5+[w^4w^2w^21]^{15}+[w^4w^510]^{15}+[w^5w^2w1]^{15}$ 5 $w$ 2 $[65,4,56]_8$ $[w^51w^4w^5]^5+[w^4w^4w^31]^{15}+[w^6w^210]^{15}+[w^2110]^{15}+[1w^4w^21]^{15}$ 5 $w$ 2 $[25,4,19]_9$ $[\beta^4\beta^31\beta^7]^{5}+[1\beta^5\beta^71]^{10}+[\beta^3\beta^6\beta^31]^{10}$ 5 $\beta$ 3 $[42,4,34]_9$ $[10\beta^20]^{6}+[\beta^71\beta 1]^{12}+[\beta^5\beta^7\beta 1]^{12}+[0\beta^4\beta^21]^{12}$ 6 $\beta^2$ 3
 $[n,k,d]_q$ Generator Matrix $N$ $\alpha$ $p^t$ $[25,4,17]_4$ $[1a^21a^2]^5+[a^2a^2a^21]^{10}+[aa^210]^{10}$ 5 $a$ 2 $[35,4,24]_4$ $[1a1a]^5+[0a^2a1]^{10}+[10a^21]^{10}+[1a^201]^{10}$ 5 $a^2$ 2 $[40,4,28]_4$ $[1a^21a^2]^5+[0a^2a1]^{10}+[a^2a^201]^{10}+[1a^201]^{10}+[1010]^5$ 5 $a$ 2 $[45,4,32]_4$ $[1a1a]^5+[a100]^{10}+[1a^211]^{10}+[0a01]^{10}+[a^2a^2a1]^{10}$ 5 $a^2$ 2 $[50,4,36]_4$ $[1a1a]^5+[1aa^21]^{10}+[1101]^{10}+[aaa^21]^{10}+[a^2a^2a1]^{10}+[a^2100]^5$ 5 $a^2$ 2 $[60,4,44]_4$ $[1a^21a^2]^5+[1110]^{10}+[0aa^21]^{10}+[00a^21]^{10}+[a^2111]^{10} +[a^2a^211]^{10}+[a0a1]^5$ 5 $a$ 2 $[65,4,48]_4$ $[1a1a]^5+[aa01]^{10}+[1011]^{10}+[a1a^21]^{10} +[1a^210]^{10}+[0a11]^{10}+[a^2100]^5+[a^20a^21]^{5}$ 5 $a^2$ 2 $[85,4,64]_4$ $[1a1a]^5+[a^2a11]^{10}+[0011]^{10}+[a^2a^201]^{10}+[1a^201]^{10}+[0a^2a^21]^{10}+[a^21a1]^{10}+[1a^2a^21]^{10}+[1010]^{5}+[0a10]^5$ 5 $a^2$ 2 $[12, 5, 6]_4$ $[a^21a^21a^2]^{6}+[1a^2a^2aa^2]^{6}$ 6 1 2 $[18, 5, 10]_4$ $[a^21a^21a^2]^{6}+[01a^201]^{6}+[a^20a^2aa]^{6}$ 6 1 2 $[20, 5, 12]_4$ $[aa110]^{10}+[a^20a^2aa]^{10}$ 10 1 2 $[30, 5,20]_4$ $[11a^2a1]^{10}+[a^2aaa^21]^{10}+[01a^201]^{10}$ 10 1 2 $[32, 5, 21]_4$ $[a^2aa^2a^20]^{8}+[aa1a^20]^{8}+[0a^20a^2a]^{8}+[1001a^2]^{8}$ 8 1 2 $[36, 5, 24]_4$ $[a^2aa^20a^2]^{12}+[a^2aaa^21]^{12}+[01a^201]^{12}$ 12 1 2 $[48, 5, 33]_4$ $[1101a^2]^{12}+[a^20a^2aa]^{12}+[0a^20a^2a]^{12}+[a^2a^2a^21a]^{12}$ 12 1 2 $[120, 5, 88]_4$ $[a^21a1a^2]^{30}+[a^2aaa^21]^{30}+[a^20a^2aa]^{30}+[0a^200a]^{30}$ 30 1 2 $[248, 5,184]_4$ $[aa^2a^2aa]^{62}+[a^2aaa^21]^{62}+[aa1a^20]^{62}+[0a^200a]^{62}$ 62 1 2 $[21,6,12]_4$ $[1a^21a^21a^2]^7+[aa0a11]^{14}$ $7$ $a$ 2 $[49,6,32]_4$ $[1a^21a^21a^2]^7+[aa^2a^2101]^{14}+[a^21aaa1]^{14}+[aa^21a^210]^{14}$ 7 $a$ 2 $[34,3,28]_8$ $[1w^6w^4]^4+[w^6w1]^{12}+[w^510]^{12}+[111]^6$ 4 $w$ 2 $[50,4,41]_8$ $[w^51w^4w^5]^5+[w^4w^2w^21]^{15}+[w^4w^510]^{15}+[w^5w^2w1]^{15}$ 5 $w$ 2 $[65,4,56]_8$ $[w^51w^4w^5]^5+[w^4w^4w^31]^{15}+[w^6w^210]^{15}+[w^2110]^{15}+[1w^4w^21]^{15}$ 5 $w$ 2 $[25,4,19]_9$ $[\beta^4\beta^31\beta^7]^{5}+[1\beta^5\beta^71]^{10}+[\beta^3\beta^6\beta^31]^{10}$ 5 $\beta$ 3 $[42,4,34]_9$ $[10\beta^20]^{6}+[\beta^71\beta 1]^{12}+[\beta^5\beta^7\beta 1]^{12}+[0\beta^4\beta^21]^{12}$ 6 $\beta^2$ 3
Some $1$-generator $SQC$ codes for $q = 4$ and $k = 5$ with BKLC parameters
 $[n,k,d]_q$ Generator Matrix $N$ $\alpha$ $p^t$ $[12,5,6]_4$ $[a^21a^21a^2]^{6}+[01111]^{6}$ 6 $1$ 2 $[20,5,12]_4$ $[aa110]^{10}+[01011]^{10}$ 10 $1$ 2 $[30,5,20]_4$ $[aa110]^{10}+[11011]^{10}+[10011]^{10}$ 10 $1$ 2 $[36,5,24]_4$ $[1101a^2]^{12}+[01011]^{12}+[10001]^{12}$ 12 $1$ 2 $[48,5,33]_4$ $[10a^2a0]^{12}+[11010]^{12}+[00111]^{12}+[10111]^{12}$ 12 $1$ 2 $[90,5,64]_4$ $[a^210a1]^{30}+[01011]^{30}+[00111]^{30}$ 30 $1$ 2 $[120,5,88]_4$ $[a^20aa^2a]^{30}+[00011]^{30}+[10111]^{30}+[10010]^{30}$ 30 $1$ 2 $[248,5,184]_4$ $[10a^210]^{62}+[00011]^{62}+[11010]^{62}+[11101]^{62}$ 62 $1$ 2
 $[n,k,d]_q$ Generator Matrix $N$ $\alpha$ $p^t$ $[12,5,6]_4$ $[a^21a^21a^2]^{6}+[01111]^{6}$ 6 $1$ 2 $[20,5,12]_4$ $[aa110]^{10}+[01011]^{10}$ 10 $1$ 2 $[30,5,20]_4$ $[aa110]^{10}+[11011]^{10}+[10011]^{10}$ 10 $1$ 2 $[36,5,24]_4$ $[1101a^2]^{12}+[01011]^{12}+[10001]^{12}$ 12 $1$ 2 $[48,5,33]_4$ $[10a^2a0]^{12}+[11010]^{12}+[00111]^{12}+[10111]^{12}$ 12 $1$ 2 $[90,5,64]_4$ $[a^210a1]^{30}+[01011]^{30}+[00111]^{30}$ 30 $1$ 2 $[120,5,88]_4$ $[a^20aa^2a]^{30}+[00011]^{30}+[10111]^{30}+[10010]^{30}$ 30 $1$ 2 $[248,5,184]_4$ $[10a^210]^{62}+[00011]^{62}+[11010]^{62}+[11101]^{62}$ 62 $1$ 2
Existence of some MDS skew $(\alpha,\theta)$-cyclic $[n,k]_q$-code
 $q$ $n$ $k$ $\alpha$ $p^t$ for $\theta$ Polynomial $x^2+ax+b$ $8$ $6$ $3, 4$ $1,1$ $2$ $x^2+x+w^3$ $9$ $6$ $3, 4$ $1,w^4$ $3$ $x^2+x+w^2$ $16$ $8$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w$ $16$ $12$ $3, 4, 5$ $1,1,1$ $2$ $x^2+wx+w$ $25$ $10$ $3, 4, 5$ $w^{12},w^6,1$ $5$ $x^2+x+w^2$ $32$ $10$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w$ $32$ $15$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w^7$ $49$ $14$ $3, 4, 5$ $w^{32},w^{24},w^{16}$ $7$ $x^2+x+w^2$ $64$ $12$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w$ $64$ $18$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w^3$
 $q$ $n$ $k$ $\alpha$ $p^t$ for $\theta$ Polynomial $x^2+ax+b$ $8$ $6$ $3, 4$ $1,1$ $2$ $x^2+x+w^3$ $9$ $6$ $3, 4$ $1,w^4$ $3$ $x^2+x+w^2$ $16$ $8$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w$ $16$ $12$ $3, 4, 5$ $1,1,1$ $2$ $x^2+wx+w$ $25$ $10$ $3, 4, 5$ $w^{12},w^6,1$ $5$ $x^2+x+w^2$ $32$ $10$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w$ $32$ $15$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w^7$ $49$ $14$ $3, 4, 5$ $w^{32},w^{24},w^{16}$ $7$ $x^2+x+w^2$ $64$ $12$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w$ $64$ $18$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w^3$
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