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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.  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

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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