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

On the dual codes of skew constacyclic codes

Alexis Eduardo Almendras Valdebenito 1, and  Andrea Luigi Tironi 2,,

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.

Keywords: Finite fields, constacyclic codes, dual codes, skew polynomial rings, semi-linear maps.
Mathematics Subject Classification: Primary: 12Y05, 16Z05; Secondary: 94B05, 94B35.
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.

[2]

M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

M'Hammed Boulagouaz and A. Leroy, (σ, δ)-codes, Adv. Math. Commun., 7 (2013), 463-474.  doi: 10.3934/amc.2013.7.463.

[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.

[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.

[14]

J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, Claredon Press - Oxford, 1979.

[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.

[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.

[17]

A. Leroy, Noncommutative polynomial maps, J. Algebra Appl., 11 (2012), 1250076, 16 pp. doi: 10.1142/S0219498812500764.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

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.

[2]

M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

M'Hammed Boulagouaz and A. Leroy, (σ, δ)-codes, Adv. Math. Commun., 7 (2013), 463-474.  doi: 10.3934/amc.2013.7.463.

[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.

[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.

[14]

J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, Claredon Press - Oxford, 1979.

[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.

[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.

[17]

A. Leroy, Noncommutative polynomial maps, J. Algebra Appl., 11 (2012), 1250076, 16 pp. doi: 10.1142/S0219498812500764.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

Table 1.  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
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$[n,k,d]_q$ 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
Table 2.  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
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$[n,k,d]_q$ 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
Table 3.  Existence of some MDS skew $(\alpha,\theta)$-cyclic $[n,k]_q$-code
$q$ $n$ $k$ $\alpha$ $p^t$ for $\theta$ Polynomial $x^2+ax+b$
$8$ $6$ $3, 4$ $1,1$ $2$ $x^2+x+w^3$
$9$ $6$ $3, 4$ $1,w^4$ $3$ $x^2+x+w^2$
$16$ $8$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w$
$16$ $12$ $3, 4, 5$ $1,1,1$ $2$ $x^2+wx+w$
$25$ $10$ $3, 4, 5$ $w^{12},w^6,1$ $5$ $x^2+x+w^2$
$32$ $10$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w$
$32$ $15$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w^7$
$49$ $14$ $3, 4, 5$ $w^{32},w^{24},w^{16}$ $7$ $x^2+x+w^2$
$64$ $12$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w$
$64$ $18$ $3, 4, 5$ $1,1,1$ $2$ $x^2+x+w^3$
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$q$ $n$ $k$ $\alpha$ $p^t$ 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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