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August  2018, 12(3): 553-577. doi: 10.3934/amc.2018033

A first step towards the skew duadic codes

Delphine Boucher

Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Received  August 2017 Revised  February 2018 Published  July 2018

Fund Project: The author is supported by the French government Investissements d’Avenir program ANR-11-LABX-0020-01.

This text gives a first definition of the $θ$-duadic codes where $θ$ is an automorphism of $\mathbb{F}_q$. A link with the self-orthogonal $θ$-cyclic codes is established. A construction and an enumeration are provided when $q$ is the square of a prime number $p$. In addition, new self-dual binary codes $ [72, 36, 12] $ are obtained from extended $θ$-duadic codes defined on $\mathbb{F}_4$.

Keywords: Finite fields, skew polynomial rings, error-correcting codes.
Mathematics Subject Classification: 94B05, 12Y05, 68W30, 12E15.
Citation: Delphine Boucher. A first step towards the skew duadic codes. Advances in Mathematics of Communications, 2018, 12 (3) : 553-577. doi: 10.3934/amc.2018033
References:
[1]

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.  Google Scholar

[2]

D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comp., 44 (2009), 1644-1656.  doi: 10.1016/j.jsc.2007.11.008.  Google Scholar

[3]

D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, Cryptography and coding, Lecture Notes in Comput. Sci., 7089 (2011), 230-243.  doi: 10.1007/978-3-642-25516-8_14.  Google Scholar

[4]

D. Boucher and F. Ulmer, Self-dual skew codes and factorization of skew polynomials, J. Symb. Comp., 60 (2014), 47-61.  doi: 10.1016/j.jsc.2013.10.003.  Google Scholar

[5]

D. Boucher, Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$, Adv. Math. Commun., 10 (2016), 765-795.  doi: 10.3934/amc.2016040.  Google Scholar

[6]

X. Caruso and J. Le Borgne, A new faster algorithm for factoring skew polynomials over finite fields, J. Symb. Comp., 79 (2017), 411-443.  doi: 10.1016/j.jsc.2016.02.016.  Google Scholar

[7]

S. T. Dougherty, T. A. Gulliver and H. Masaaki, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.   Google Scholar

[8]

M. Giesbrecht, Factoring in skew-polynomial rings over finite fields, J. Symb. Comput., 26 (1998), 463-486.  doi: 10.1006/jsco.1998.0224.  Google Scholar

[9]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.  Google Scholar

[10]

N. Jacobson, The Theory of Rings, Amer. Math. Soc., 1943.  Google Scholar

[11]

A. Kaya, B. Yildiz and I. Siap, New extremal binary self-dual codes of length 68 from quadratic residue codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$, Finite Fields and their Applications, 29 (2014), 160-177.  doi: 10.1016/j.ffa.2014.04.009.  Google Scholar

[12]

R. W. K. Odoni, On additive polynomials over a finite field, Proc. Edinburgh Math. Soc., 42 (1999), 1-16.  doi: 10.1017/S0013091500019970.  Google Scholar

[13]

O. Ore, Theory of Non-Commutative Polynomials, Ann. Math., 34 (1933), 480-508.  doi: 10.2307/1968173.  Google Scholar

[14]

J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9781139856065.  Google Scholar

[15]

A. Zhdanov, New self-dual codes of length $72$, preprint, arXiv: 1705.05779. Google Scholar

show all references

References:
[1]

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.  Google Scholar

[2]

D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comp., 44 (2009), 1644-1656.  doi: 10.1016/j.jsc.2007.11.008.  Google Scholar

[3]

D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, Cryptography and coding, Lecture Notes in Comput. Sci., 7089 (2011), 230-243.  doi: 10.1007/978-3-642-25516-8_14.  Google Scholar

[4]

D. Boucher and F. Ulmer, Self-dual skew codes and factorization of skew polynomials, J. Symb. Comp., 60 (2014), 47-61.  doi: 10.1016/j.jsc.2013.10.003.  Google Scholar

[5]

D. Boucher, Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$, Adv. Math. Commun., 10 (2016), 765-795.  doi: 10.3934/amc.2016040.  Google Scholar

[6]

X. Caruso and J. Le Borgne, A new faster algorithm for factoring skew polynomials over finite fields, J. Symb. Comp., 79 (2017), 411-443.  doi: 10.1016/j.jsc.2016.02.016.  Google Scholar

[7]

S. T. Dougherty, T. A. Gulliver and H. Masaaki, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.   Google Scholar

[8]

M. Giesbrecht, Factoring in skew-polynomial rings over finite fields, J. Symb. Comput., 26 (1998), 463-486.  doi: 10.1006/jsco.1998.0224.  Google Scholar

[9]

W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.  Google Scholar

[10]

N. Jacobson, The Theory of Rings, Amer. Math. Soc., 1943.  Google Scholar

[11]

A. Kaya, B. Yildiz and I. Siap, New extremal binary self-dual codes of length 68 from quadratic residue codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2+u^2{\mathbb{F}}_2$, Finite Fields and their Applications, 29 (2014), 160-177.  doi: 10.1016/j.ffa.2014.04.009.  Google Scholar

[12]

R. W. K. Odoni, On additive polynomials over a finite field, Proc. Edinburgh Math. Soc., 42 (1999), 1-16.  doi: 10.1017/S0013091500019970.  Google Scholar

[13]

O. Ore, Theory of Non-Commutative Polynomials, Ann. Math., 34 (1933), 480-508.  doi: 10.2307/1968173.  Google Scholar

[14]

J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9781139856065.  Google Scholar

[15]

A. Zhdanov, New self-dual codes of length $72$, preprint, arXiv: 1705.05779. Google Scholar

Table 5.  Type Ⅱ $[72, 36, 12]$ self-dual codes who are binary images of $[36, 18]_4$ self-dual extended $\theta$-cyclic codes
Coefficients of $g$ $v$ $ \alpha$
$\left[ a, a, 0, a, a^2, a, a, 0, 0, 0, 1, 1, a^2, 1, 0, 1, 1 \right]$ $\left[1, a, 1, a^2\right]$ -3072
$\left[ a, a^2, 0, a^2, a^2, a^2, 0, a^2, 0, a^2, 0, a^2, a^2, a^2, 0, a^2, 1 \right]$ $\left[ 1, a, 1, a^2 \right]$ -3276
$\left[ a^2, 1, a^2, a^2, 1, a^2, 0, 1, 0, a^2, 0, 1, a^2, 1, 1, a^2, 1 \right]$ $\left[ 1, a^2, 1, a \right]$ -3480
$\left[ a, 1, a, 1, 0, 0, a, a^2, 0, a^2, 1, 0, 0, a, 1, a, 1 \right]$ $\left[1, a, 1, a^2\right]$ -3582
$\left[ a, 0, a^2, 0, 0, 1, 1, a^2, 0, a^2, a, a, 0, 0, a^2, 0, 1 \right]$ $\left[1, a, 1, a^2\right]$ -3684
$\left[ a^2, 1, 0, a, 0, 1, a^2, a, a, a, 1, a^2, 0, a, 0, a^2, 1 \right]$ $\left[ 1, a^2, 1, a \right]$ -3990
$\left[ a, a^2, a, 0, 0, 1, a, 1, 0, a, 1, a, 0, 0, 1, a^2, 1 \right]$ $\left[1, a, 1, a^2\right]$ -4092
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Coefficients of $g$ $v$ $ \alpha$
$\left[ a, a, 0, a, a^2, a, a, 0, 0, 0, 1, 1, a^2, 1, 0, 1, 1 \right]$ $\left[1, a, 1, a^2\right]$ -3072
$\left[ a, a^2, 0, a^2, a^2, a^2, 0, a^2, 0, a^2, 0, a^2, a^2, a^2, 0, a^2, 1 \right]$ $\left[ 1, a, 1, a^2 \right]$ -3276
$\left[ a^2, 1, a^2, a^2, 1, a^2, 0, 1, 0, a^2, 0, 1, a^2, 1, 1, a^2, 1 \right]$ $\left[ 1, a^2, 1, a \right]$ -3480
$\left[ a, 1, a, 1, 0, 0, a, a^2, 0, a^2, 1, 0, 0, a, 1, a, 1 \right]$ $\left[1, a, 1, a^2\right]$ -3582
$\left[ a, 0, a^2, 0, 0, 1, 1, a^2, 0, a^2, a, a, 0, 0, a^2, 0, 1 \right]$ $\left[1, a, 1, a^2\right]$ -3684
$\left[ a^2, 1, 0, a, 0, 1, a^2, a, a, a, 1, a^2, 0, a, 0, a^2, 1 \right]$ $\left[ 1, a^2, 1, a \right]$ -3990
$\left[ a, a^2, a, 0, 0, 1, a, 1, 0, a, 1, a, 0, 0, 1, a^2, 1 \right]$ $\left[1, a, 1, a^2\right]$ -4092
Algorithm 1.  Irreducible skew polynomials of $\mathbb{F}_{p^2}[X; \theta]$ with a given bound
Require: $f \in \mathbb{F}_p[X^2]$ irreducible in $\mathbb{F}_p[X^2]$
Ensure: All irreducible skew polynomials with bound $f(X^2)$
1. $d \leftarrow \deg_{X^2} f(X^2)$
2. $\alpha \leftarrow$ root of $f$ in $\overline{\mathbb{F}_p}$
3. if $d$ is odd then
4.    $\delta \leftarrow (d-1)/2$
5.    $E \leftarrow \emptyset$
6.    for $u \in \mathbb{F}_{p^{2d}}$ such that $u^{p^d+1}=\alpha$ do
7.      $P \leftarrow$ Interpolation Polynomial in $\mathbb{F}_{p^2}[Z]$ at the $d$ points $[\alpha^{p^{2i}}, u^{p^{2i}}]_{0 \leq i \leq \delta}$ and $ [\alpha^{p^{2i+1}}, \alpha^{p^{2i+1}}/u^{p^{2i+1}}]_{0 \leq i \leq \delta-1}$
8.      $(A, B) \leftarrow$ solution of the Cauchy interpolation problem ${\displaystyle \frac{A}{B} \equiv P \pmod{f}}$ with $B$ monic, $\deg(B)=\delta$, $\deg(A) \leq \delta$
9.      add $A(X^2) + X \cdot B(X^2)$ to the set $E$
10.    endfor
11. else
12.    $\delta \leftarrow d/2$
13.    $E \leftarrow \{\tilde{f}(X^2), \Theta(\tilde{f})(X^2)\}$ where $\tilde{f}(Z) \Theta(\tilde{f})(Z)=f(Z)$ is the factorization of $f(Z)$ in $\mathbb{F}_{p^2}[Z]$
14.    for $u \in \mathbb{F}_{p^{d}}$ such that $u \neq 0$ do
15.      $P \leftarrow$ Interpolation Polynomial in $\mathbb{F}_{p^2}[Z]$ at the $d$ points $[\alpha^{p^{2i}}, u^{p^{2i}}]_{0 \leq i \leq \delta-1} $ and $[\alpha^{p^{2i+1}}, \alpha^{p^{2i+1}}/u^{p^{2i+1}}]_{0 \leq i \leq \delta-1}$
16.      $(A, B) \leftarrow$ solution of the Cauchy interpolation problem ${\displaystyle \frac{A}{B} \equiv P \pmod{f}}$ with $A$ monic, $\deg(A)=\delta$, $\deg(B)<\delta$
17.      add $A(X^2) + X \cdot B(X^2)$ to the set $E$
18.    endfor
19. endif
20. return $E$
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Require: $f \in \mathbb{F}_p[X^2]$ irreducible in $\mathbb{F}_p[X^2]$
Ensure: All irreducible skew polynomials with bound $f(X^2)$
1. $d \leftarrow \deg_{X^2} f(X^2)$
2. $\alpha \leftarrow$ root of $f$ in $\overline{\mathbb{F}_p}$
3. if $d$ is odd then
4.    $\delta \leftarrow (d-1)/2$
5.    $E \leftarrow \emptyset$
6.    for $u \in \mathbb{F}_{p^{2d}}$ such that $u^{p^d+1}=\alpha$ do
7.      $P \leftarrow$ Interpolation Polynomial in $\mathbb{F}_{p^2}[Z]$ at the $d$ points $[\alpha^{p^{2i}}, u^{p^{2i}}]_{0 \leq i \leq \delta}$ and $ [\alpha^{p^{2i+1}}, \alpha^{p^{2i+1}}/u^{p^{2i+1}}]_{0 \leq i \leq \delta-1}$
8.      $(A, B) \leftarrow$ solution of the Cauchy interpolation problem ${\displaystyle \frac{A}{B} \equiv P \pmod{f}}$ with $B$ monic, $\deg(B)=\delta$, $\deg(A) \leq \delta$
9.      add $A(X^2) + X \cdot B(X^2)$ to the set $E$
10.    endfor
11. else
12.    $\delta \leftarrow d/2$
13.    $E \leftarrow \{\tilde{f}(X^2), \Theta(\tilde{f})(X^2)\}$ where $\tilde{f}(Z) \Theta(\tilde{f})(Z)=f(Z)$ is the factorization of $f(Z)$ in $\mathbb{F}_{p^2}[Z]$
14.    for $u \in \mathbb{F}_{p^{d}}$ such that $u \neq 0$ do
15.      $P \leftarrow$ Interpolation Polynomial in $\mathbb{F}_{p^2}[Z]$ at the $d$ points $[\alpha^{p^{2i}}, u^{p^{2i}}]_{0 \leq i \leq \delta-1} $ and $[\alpha^{p^{2i+1}}, \alpha^{p^{2i+1}}/u^{p^{2i+1}}]_{0 \leq i \leq \delta-1}$
16.      $(A, B) \leftarrow$ solution of the Cauchy interpolation problem ${\displaystyle \frac{A}{B} \equiv P \pmod{f}}$ with $A$ monic, $\deg(A)=\delta$, $\deg(B)<\delta$
17.      add $A(X^2) + X \cdot B(X^2)$ to the set $E$
18.    endfor
19. endif
20. return $E$
Table 1.  Parametrization of the irreducible monic skew polynomials of $\mathbb{F}_4[X;\theta]$ bounded by $X^6+X^2+1$.
$u$ $P(Z) \in \mathbb{F}_4[Z]$ $h(X) \in \mathbb{F}_4[X;\theta]$
$\gamma$ $Z^2 + a^2 \, Z + 1$ $X^3 + a \, X^2 + a \, X + a$
$\gamma^8$ $Z^2 + a \, Z + 1$ $X^3 + a^2 \, X^2 + a^2 \, X + a^2$
$\gamma^{15}$ $a^2 \, Z^2 + a^2 \, Z$ $X^3 + X + a^2$
$\gamma^{22}$ $a \, Z^2 + Z + a$ $X^3 + a^2 \, X^2 + a \, X + a^2$
$\gamma^{29}$ $a \, Z^2 + a^2 \, Z + a$ $X^3 + X^2 + a^2 \, X + 1$
$\gamma^{36}$ $Z^2 + Z$ $X^3 + X + 1$
$\gamma^{43}$ $a^2 \, Z^2 + a \, Z + a^2$ $X^3 + X^2 + a \, X + 1$
$\gamma^{50}$ $a^2 \, Z^2 + Z + a^2$ $X^3 + a \, X^2 + a^2 \, X + a$
$\gamma^{57}$ $a \, Z^2 + a \, Z$ $X^3 + X + a$
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$u$ $P(Z) \in \mathbb{F}_4[Z]$ $h(X) \in \mathbb{F}_4[X;\theta]$
$\gamma$ $Z^2 + a^2 \, Z + 1$ $X^3 + a \, X^2 + a \, X + a$
$\gamma^8$ $Z^2 + a \, Z + 1$ $X^3 + a^2 \, X^2 + a^2 \, X + a^2$
$\gamma^{15}$ $a^2 \, Z^2 + a^2 \, Z$ $X^3 + X + a^2$
$\gamma^{22}$ $a \, Z^2 + Z + a$ $X^3 + a^2 \, X^2 + a \, X + a^2$
$\gamma^{29}$ $a \, Z^2 + a^2 \, Z + a$ $X^3 + X^2 + a^2 \, X + 1$
$\gamma^{36}$ $Z^2 + Z$ $X^3 + X + 1$
$\gamma^{43}$ $a^2 \, Z^2 + a \, Z + a^2$ $X^3 + X^2 + a \, X + 1$
$\gamma^{50}$ $a^2 \, Z^2 + Z + a^2$ $X^3 + a \, X^2 + a^2 \, X + a$
$\gamma^{57}$ $a \, Z^2 + a \, Z$ $X^3 + X + a$
Table 2.  Parametrization of the irreducible monic skew polynomials of $\mathbb{F}_4[X;\theta]$ bounded by $X^8+X^2+1$ and distinct of $X^4+X^2+a$ and $X^4+X^2+a^2$.
$u$ $P(Z) \in \mathbb{F}_4[Z]$ $h(X) \in \mathbb{F}_4[X;\theta]$
$1$ $Z^3 + a^2 \, Z + a^2$ $X^4 + a^2 \, X^3 + a^2 \, X^2 + a$
$\alpha$ $Z^3 + a \, Z + a$ $X^4 + a \, X^3 + a \, X^2 + a^2$
$\alpha^2$ $a^2 \, Z^3 + a^2 \, Z^2 + a$ $X^4 + a^2 \, X^3 + a \, X^2 + a^2 \, X + a$
$\alpha^3$ $a^2 \, Z^2 + a^2$ $X^4 + a^2 \, X + 1$
$\alpha^4$ $a^2 \, Z^3 + a^2 \, Z^2 + 1$ $X^4 + a^2 \, X^3 + a^2 \, X^2 + a^2 \, X + a^2$
$\alpha^5$ $a \, Z^3 + Z + 1$ $X^4 + X^3 + a^2 \, X^2 + a$
$\alpha^6$ $a \, Z^3 + a^2 \, Z + a^2$ $X^4 + a^2 \, X^3 + a \, X^2 + a^2$
$\alpha^7$ $Z^3 + Z^2 + a^2$ $X^4 + X^3 + a \, X^2 + X + a$
$\alpha^8$ $Z^2 + 1$ $X^4 + X + 1$
$\alpha^9$ $Z^3 + Z^2 + a$ $X^4 + X^3 + a^2 \, X^2 + X + a^2$
$\alpha^{10}$ $a^2 \, Z^3 + a \, Z + a$ $X^4 + a \, X^3 + a^2 \, X^2 + a$
$\alpha^{11}$ $a^2 \, Z^3 + Z + 1$ $X^4 + X^3 + a \, X^2 + a^2$
$\alpha^{12}$ $a \, Z^3 + a \, Z^2 + 1$ $X^4 + a \, X^3 + a \, X^2 + a \, X + a$
$\alpha^{13}$ $a \, Z^2 + a$ $X^4 + a \, X + 1$
$\alpha^{14}$ $a \, Z^3 + a \, Z^2 + a^2$ $X^4 + a \, X^3 + a^2 \, X^2 + a \, X + a^2$
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$u$ $P(Z) \in \mathbb{F}_4[Z]$ $h(X) \in \mathbb{F}_4[X;\theta]$
$1$ $Z^3 + a^2 \, Z + a^2$ $X^4 + a^2 \, X^3 + a^2 \, X^2 + a$
$\alpha$ $Z^3 + a \, Z + a$ $X^4 + a \, X^3 + a \, X^2 + a^2$
$\alpha^2$ $a^2 \, Z^3 + a^2 \, Z^2 + a$ $X^4 + a^2 \, X^3 + a \, X^2 + a^2 \, X + a$
$\alpha^3$ $a^2 \, Z^2 + a^2$ $X^4 + a^2 \, X + 1$
$\alpha^4$ $a^2 \, Z^3 + a^2 \, Z^2 + 1$ $X^4 + a^2 \, X^3 + a^2 \, X^2 + a^2 \, X + a^2$
$\alpha^5$ $a \, Z^3 + Z + 1$ $X^4 + X^3 + a^2 \, X^2 + a$
$\alpha^6$ $a \, Z^3 + a^2 \, Z + a^2$ $X^4 + a^2 \, X^3 + a \, X^2 + a^2$
$\alpha^7$ $Z^3 + Z^2 + a^2$ $X^4 + X^3 + a \, X^2 + X + a$
$\alpha^8$ $Z^2 + 1$ $X^4 + X + 1$
$\alpha^9$ $Z^3 + Z^2 + a$ $X^4 + X^3 + a^2 \, X^2 + X + a^2$
$\alpha^{10}$ $a^2 \, Z^3 + a \, Z + a$ $X^4 + a \, X^3 + a^2 \, X^2 + a$
$\alpha^{11}$ $a^2 \, Z^3 + Z + 1$ $X^4 + X^3 + a \, X^2 + a^2$
$\alpha^{12}$ $a \, Z^3 + a \, Z^2 + 1$ $X^4 + a \, X^3 + a \, X^2 + a \, X + a$
$\alpha^{13}$ $a \, Z^2 + a$ $X^4 + a \, X + 1$
$\alpha^{14}$ $a \, Z^3 + a \, Z^2 + a^2$ $X^4 + a \, X^3 + a^2 \, X^2 + a \, X + a^2$
Table 3.  Type Ⅱ $[72, 36, 12]$ self-dual codes who are binary images of $[36, 18]_4$ self-dual $\theta$-cyclic codes
Coefficients of $g$ $\alpha$
$\left[ a^2, 0, a^2, a^2, 1, 1, a^2, 1, a^2, 0, 1, a^2, 1, a^2, a^2, 1, 1, 0, 1 \right]$ -2820
$\left[ 1, a, a, a, a^2, a^2, a, 0, 0, 0, 0, 0, a^2, a, a, a^2, a^2, a^2, 1 \right]$ -3204
$\left[ 1, a^2, a^2, 1, 1, a^2, 1, a, 0, 0, 0, a^2, 1, a, 1, 1, a, a, 1 \right]$ -3276
$\left[ a^2, 1, 1, 0, a, 1, 1, a^2, 0, 0, 0, 1, a^2, a^2, a, 0, a^2, a^2, 1 \right]$ -3312
$\left[ a^2, a^2, 1, a, a^2, 0, 0, 0, a, 0, a, 0, 0, 0, 1, a, a^2, 1, 1 \right]$ -3336
$\left[ a^2, a, a, 0, 1, a, a, a^2, a^2, 0, 1, 1, a, a, a^2, 0, a, a, 1 \right]$ -3372
$\left[ a^2, 0, 0, a^2, 0, a, a, a^2, a, a, a, 1, a, a, 0, 1, 0, 0, 1 \right]$ -3408
$\left[ 1, 1, a, a^2, 1, 1, 1, a, 0, 1, 0, a^2, 1, 1, 1, a, a^2, 1, 1 \right]$ -3420
$\left[ a, a, 1, a, a^2, a^2, 1, a^2, a^2, a^2, a^2, a^2, a, a^2, a^2, 1, a, 1, 1 \right]$ -3456
$\left[ a, a^2, 1, a, 0, a, 0, a^2, a, a^2, 1, a^2, 0, 1, 0, 1, a, a^2, 1 \right]$ -3504
$\left[ 1, a, a^2, a, a^2, 1, 1, a, a, 0, a^2, a^2, 1, 1, a, a^2, a, a^2, 1 \right]$ -3540
$\left[ a, 1, a^2, a^2, a, 0, a, 0, 0, 0, 0, 0, 1, 0, 1, a^2, a^2, a, 1 \right]$ -3564
$\left[ 1, 0, 0, a, 1, 1, a^2, a, 0, 1, 0, a^2, a, 1, 1, a^2, 0, 0, 1 \right]$ -3576
$\left[ 1, 1, a^2, a^2, 1, a^2, a, a^2, a, 0, a^2, a, a^2, a, 1, a, a, 1, 1 \right]$ -3600
$\left[ 1, 0, 0, 0, 1, 1, 1, 0, a, 1, a^2, 0, 1, 1, 1, 0, 0, 0, 1 \right]$ -3612
$\left[ 1, 0, 0, 0, 1, a^2, 0, 1, a^2, 0, a, 1, 0, a, 1, 0, 0, 0, 1 \right]$ -3636
$\left[ a, a^2, a^2, a^2, 1, 1, a^2, 0, a, 0, 1, 0, a^2, a, a, a^2, a^2, a^2, 1 \right]$ -3660
$\left[ 1, 0, 0, a, 0, a, a, 1, 1, 1, 1, 1, a^2, a^2, 0, a^2, 0, 0, 1 \right]$ -3696
$\left[ a, 0, 0, a, a, 1, a^2, a^2, a, a^2, 1, a^2, a^2, a, 1, 1, 0, 0, 1 \right]$ -3732
$\left[ a, 0, a, a, 1, a^2, 0, a^2, 0, 0, 0, a^2, 0, a^2, a, 1, 1, 0, 1 \right]$ -3744
$\left[ a^2, a, 1, 1, a^2, a^2, 1, 0, a^2, a, 1, 0, a^2, 1, 1, a^2, a^2, a, 1 \right]$ -3768
$\left[ 1, 1, a^2, 0, a, 0, a, 1, 0, 1, 0, 1, a^2, 0, a^2, 0, a, 1, 1 \right]$ -3816
$\left[ 1, a^2, a^2, a, 0, a^2, a, a, a, 1, a^2, a^2, a^2, a, 0, a^2, a, a, 1 \right]$ -3828
$\left[ 1, a, a, 1, 0, a^2, 0, a^2, 0, 0, 0, a, 0, a, 0, 1, a^2, a^2, 1 \right]$ -3924
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Coefficients of $g$ $\alpha$
$\left[ a^2, 0, a^2, a^2, 1, 1, a^2, 1, a^2, 0, 1, a^2, 1, a^2, a^2, 1, 1, 0, 1 \right]$ -2820
$\left[ 1, a, a, a, a^2, a^2, a, 0, 0, 0, 0, 0, a^2, a, a, a^2, a^2, a^2, 1 \right]$ -3204
$\left[ 1, a^2, a^2, 1, 1, a^2, 1, a, 0, 0, 0, a^2, 1, a, 1, 1, a, a, 1 \right]$ -3276
$\left[ a^2, 1, 1, 0, a, 1, 1, a^2, 0, 0, 0, 1, a^2, a^2, a, 0, a^2, a^2, 1 \right]$ -3312
$\left[ a^2, a^2, 1, a, a^2, 0, 0, 0, a, 0, a, 0, 0, 0, 1, a, a^2, 1, 1 \right]$ -3336
$\left[ a^2, a, a, 0, 1, a, a, a^2, a^2, 0, 1, 1, a, a, a^2, 0, a, a, 1 \right]$ -3372
$\left[ a^2, 0, 0, a^2, 0, a, a, a^2, a, a, a, 1, a, a, 0, 1, 0, 0, 1 \right]$ -3408
$\left[ 1, 1, a, a^2, 1, 1, 1, a, 0, 1, 0, a^2, 1, 1, 1, a, a^2, 1, 1 \right]$ -3420
$\left[ a, a, 1, a, a^2, a^2, 1, a^2, a^2, a^2, a^2, a^2, a, a^2, a^2, 1, a, 1, 1 \right]$ -3456
$\left[ a, a^2, 1, a, 0, a, 0, a^2, a, a^2, 1, a^2, 0, 1, 0, 1, a, a^2, 1 \right]$ -3504
$\left[ 1, a, a^2, a, a^2, 1, 1, a, a, 0, a^2, a^2, 1, 1, a, a^2, a, a^2, 1 \right]$ -3540
$\left[ a, 1, a^2, a^2, a, 0, a, 0, 0, 0, 0, 0, 1, 0, 1, a^2, a^2, a, 1 \right]$ -3564
$\left[ 1, 0, 0, a, 1, 1, a^2, a, 0, 1, 0, a^2, a, 1, 1, a^2, 0, 0, 1 \right]$ -3576
$\left[ 1, 1, a^2, a^2, 1, a^2, a, a^2, a, 0, a^2, a, a^2, a, 1, a, a, 1, 1 \right]$ -3600
$\left[ 1, 0, 0, 0, 1, 1, 1, 0, a, 1, a^2, 0, 1, 1, 1, 0, 0, 0, 1 \right]$ -3612
$\left[ 1, 0, 0, 0, 1, a^2, 0, 1, a^2, 0, a, 1, 0, a, 1, 0, 0, 0, 1 \right]$ -3636
$\left[ a, a^2, a^2, a^2, 1, 1, a^2, 0, a, 0, 1, 0, a^2, a, a, a^2, a^2, a^2, 1 \right]$ -3660
$\left[ 1, 0, 0, a, 0, a, a, 1, 1, 1, 1, 1, a^2, a^2, 0, a^2, 0, 0, 1 \right]$ -3696
$\left[ a, 0, 0, a, a, 1, a^2, a^2, a, a^2, 1, a^2, a^2, a, 1, 1, 0, 0, 1 \right]$ -3732
$\left[ a, 0, a, a, 1, a^2, 0, a^2, 0, 0, 0, a^2, 0, a^2, a, 1, 1, 0, 1 \right]$ -3744
$\left[ a^2, a, 1, 1, a^2, a^2, 1, 0, a^2, a, 1, 0, a^2, 1, 1, a^2, a^2, a, 1 \right]$ -3768
$\left[ 1, 1, a^2, 0, a, 0, a, 1, 0, 1, 0, 1, a^2, 0, a^2, 0, a, 1, 1 \right]$ -3816
$\left[ 1, a^2, a^2, a, 0, a^2, a, a, a, 1, a^2, a^2, a^2, a, 0, a^2, a, a, 1 \right]$ -3828
$\left[ 1, a, a, 1, 0, a^2, 0, a^2, 0, 0, 0, a, 0, a, 0, 1, a^2, a^2, 1 \right]$ -3924
Table 4.  Type Ⅰ $[72, 36, 12]$ self-dual codes who are binary images of $[36, 18]_4$ self-dual $\theta$-cyclic codes.
Coefficients of $g$ $\beta$ $\gamma$
$\left[ a^2, a, 1, 1, a^2, a^2, 1, 1, a, a, a, a^2, a^2, 1, 1, a^2, a^2, a, 1 \right]$ 201 0
$\left[ 1, 0, 0, a, 0, a^2, a, 0, a, 0, a^2, 0, a^2, a, 0, a^2, 0, 0, 1 \right]$ 237 0
$\left[ 1, a^2, a^2, a^2, a, 1, a^2, a, a, 1, a^2, a^2, a, 1, a^2, a, a, a, 1 \right]$ 249 0
$\left[ 1, 0, 1, a, a, 0, 1, 1, a^2, 1, a, 1, 1, 0, a^2, a^2, 1, 0, 1 \right]$ 273 0
$\left[ a, 1, 1, 1, a^2, a, 1, a^2, 1, a^2, a, a^2, a, 1, a^2, a, a, a, 1 \right]$ 273 36
$\left[ a^2, a^2, 1, 0, 1, 0, 0, a^2, 1, 0, a^2, 1, 0, 0, a^2, 0, a^2, 1, 1 \right]$ 309 0
$\left[ 1, 1, a, 1, a^2, a^2, a^2, 0, a, 1, a^2, 0, a, a, a, 1, a^2, 1, 1 \right]$ 345 0
$\left[ a^2, a, 1, a^2, 0, 0, 1, 0, a^2, a, 1, 0, a^2, 0, 0, 1, a^2, a, 1 \right]$ 381 0
$\left[ 1, a, a, a^2, 0, a, a, 1, 1, 1, 1, 1, a^2, a^2, 0, a, a^2, a^2, 1 \right]$ 393 36
$\left[ a, a, a^2, a, 1, a, 0, a, a^2, 0, a^2, 1, 0, 1, a, 1, a^2, 1, 1 \right]$ 489 36
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Coefficients of $g$ $\beta$ $\gamma$
$\left[ a^2, a, 1, 1, a^2, a^2, 1, 1, a, a, a, a^2, a^2, 1, 1, a^2, a^2, a, 1 \right]$ 201 0
$\left[ 1, 0, 0, a, 0, a^2, a, 0, a, 0, a^2, 0, a^2, a, 0, a^2, 0, 0, 1 \right]$ 237 0
$\left[ 1, a^2, a^2, a^2, a, 1, a^2, a, a, 1, a^2, a^2, a, 1, a^2, a, a, a, 1 \right]$ 249 0
$\left[ 1, 0, 1, a, a, 0, 1, 1, a^2, 1, a, 1, 1, 0, a^2, a^2, 1, 0, 1 \right]$ 273 0
$\left[ a, 1, 1, 1, a^2, a, 1, a^2, 1, a^2, a, a^2, a, 1, a^2, a, a, a, 1 \right]$ 273 36
$\left[ a^2, a^2, 1, 0, 1, 0, 0, a^2, 1, 0, a^2, 1, 0, 0, a^2, 0, a^2, 1, 1 \right]$ 309 0
$\left[ 1, 1, a, 1, a^2, a^2, a^2, 0, a, 1, a^2, 0, a, a, a, 1, a^2, 1, 1 \right]$ 345 0
$\left[ a^2, a, 1, a^2, 0, 0, 1, 0, a^2, a, 1, 0, a^2, 0, 0, 1, a^2, a, 1 \right]$ 381 0
$\left[ 1, a, a, a^2, 0, a, a, 1, 1, 1, 1, 1, a^2, a^2, 0, a, a^2, a^2, 1 \right]$ 393 36
$\left[ a, a, a^2, a, 1, a, 0, a, a^2, 0, a^2, 1, 0, 1, a, 1, a^2, 1, 1 \right]$ 489 36
Table 6.  Type Ⅰ $[72, 36, 12]$ self-dual codes who are binary images of $[36, 18]_4$ self-dual extended $\theta$-cyclic codes.
Coefficients of $g$ $v$ $ \beta$ $\delta$
$\left[ 1, 1, 0, 0, a, 0, a, 1, 1, 1, a^2, 0, a^2, 0, 0, 1, 1 \right]$ $\left[0, 1, 1, 0\right]$ 221 0
$\left[ 1, a^2, 1, 1, a, a^2, a^2, a^2, 0, a, a, a, a^2, 1, 1, a, 1 \right]$ $\left[0, 1, 1, 0\right]$ 323 0
$\left[ a, 1, a, 1, 0, 0, a, a^2, 0, a^2, 1, 0, 0, a, 1, a, 1 \right]$ $\left[0, a^2, a, 0\right]$ 238 0
$\left[ a, a, 0, a, a^2, a, a, 0, 0, 0, 1, 1, a^2, 1, 0, 1, 1 \right]$ $\left[0, a^2, a, 0\right]$ 391 0
$\left[ a, a, 0, 1, 0, 0, a, 0, a^2, 0, 1, 0, 0, a, 0, 1, 1 \right]$ $\left[0, a^2, a, 0\right]$ 289 0
$\left[ a^2, 1, 0, a, 0, 1, a^2, a, a, a, 1, a^2, 0, a, 0, a^2, 1 \right]$ $\left[0, a, a^2, 0\right]$ 102 0
$\left[ a, 0, 1, a^2, 0, a, 0, a^2, 0, a^2, 0, 1, 0, a^2, a, 0, 1 \right]$ $\left[0, a^2, a, 0\right]$ 255 0
$\left[ a, a^2, a, 0, 0, 1, a, 1, 0, a, 1, a, 0, 0, 1, a^2, 1 \right]$ $\left[0, a^2, a, 0\right]$ 153 0
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Coefficients of $g$ $v$ $ \beta$ $\delta$
$\left[ 1, 1, 0, 0, a, 0, a, 1, 1, 1, a^2, 0, a^2, 0, 0, 1, 1 \right]$ $\left[0, 1, 1, 0\right]$ 221 0
$\left[ 1, a^2, 1, 1, a, a^2, a^2, a^2, 0, a, a, a, a^2, 1, 1, a, 1 \right]$ $\left[0, 1, 1, 0\right]$ 323 0
$\left[ a, 1, a, 1, 0, 0, a, a^2, 0, a^2, 1, 0, 0, a, 1, a, 1 \right]$ $\left[0, a^2, a, 0\right]$ 238 0
$\left[ a, a, 0, a, a^2, a, a, 0, 0, 0, 1, 1, a^2, 1, 0, 1, 1 \right]$ $\left[0, a^2, a, 0\right]$ 391 0
$\left[ a, a, 0, 1, 0, 0, a, 0, a^2, 0, 1, 0, 0, a, 0, 1, 1 \right]$ $\left[0, a^2, a, 0\right]$ 289 0
$\left[ a^2, 1, 0, a, 0, 1, a^2, a, a, a, 1, a^2, 0, a, 0, a^2, 1 \right]$ $\left[0, a, a^2, 0\right]$ 102 0
$\left[ a, 0, 1, a^2, 0, a, 0, a^2, 0, a^2, 0, 1, 0, a^2, a, 0, 1 \right]$ $\left[0, a^2, a, 0\right]$ 255 0
$\left[ a, a^2, a, 0, 0, 1, a, 1, 0, a, 1, a, 0, 0, 1, a^2, 1 \right]$ $\left[0, a^2, a, 0\right]$ 153 0
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