A first step towards the skew duadic codes

Delphine Boucher

doi:10.3934/amc.2018033

Article Contents

2018, Volume 12, Issue 3: 553-577. Doi: 10.3934/amc.2018033

This issue Previous Article An asymmetric ZCZ sequence set with inter-subset uncorrelated property and flexible ZCZ length Next Article Parameters of LCD BCH codes with two lengths

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

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

Abstract / Introduction Full Text(HTML) Figure(0) / Table(7) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 94B05, 12Y05, 68W30, 12E15.

Citation:

Full Text(HTML)

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

| Show Table

DownLoad: CSV

Table 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

$E$

| Show Table

DownLoad: CSV

Table 1. Parametrization of the irreducible monic skew polynomials of $\mathbb{F}_4[X;\theta]$ bounded by $X^6+X^2+1$.

$u$	$P(Z) \in \mathbb{F}_4[Z]$	$h(X) \in \mathbb{F}_4[X;\theta]$
$\gamma$	$Z^2 + a^2 \, Z + 1$	$X^3 + a \, X^2 + a \, X + a$
$\gamma^8$	$Z^2 + a \, Z + 1$	$X^3 + a^2 \, X^2 + a^2 \, X + a^2$
$\gamma^{15}$	$a^2 \, Z^2 + a^2 \, Z$	$X^3 + X + a^2$
$\gamma^{22}$	$a \, Z^2 + Z + a$	$X^3 + a^2 \, X^2 + a \, X + a^2$
$\gamma^{29}$	$a \, Z^2 + a^2 \, Z + a$	$X^3 + X^2 + a^2 \, X + 1$
$\gamma^{36}$	$Z^2 + Z$	$X^3 + X + 1$
$\gamma^{43}$	$a^2 \, Z^2 + a \, Z + a^2$	$X^3 + X^2 + a \, X + 1$
$\gamma^{50}$	$a^2 \, Z^2 + Z + a^2$	$X^3 + a \, X^2 + a^2 \, X + a$
$\gamma^{57}$	$a \, Z^2 + a \, Z$	$X^3 + X + a$

| Show Table

DownLoad: CSV

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 $X^4+X^2+a^2$.

$u$	$P(Z) \in \mathbb{F}_4[Z]$	$h(X) \in \mathbb{F}_4[X;\theta]$
$1$	$Z^3 + a^2 \, Z + a^2$	$X^4 + a^2 \, X^3 + a^2 \, X^2 + a$
$\alpha$	$Z^3 + a \, Z + a$	$X^4 + a \, X^3 + a \, X^2 + a^2$
$\alpha^2$	$a^2 \, Z^3 + a^2 \, Z^2 + a$	$X^4 + a^2 \, X^3 + a \, X^2 + a^2 \, X + a$
$\alpha^3$	$a^2 \, Z^2 + a^2$	$X^4 + a^2 \, X + 1$
$\alpha^4$	$a^2 \, Z^3 + a^2 \, Z^2 + 1$	$X^4 + a^2 \, X^3 + a^2 \, X^2 + a^2 \, X + a^2$
$\alpha^5$	$a \, Z^3 + Z + 1$	$X^4 + X^3 + a^2 \, X^2 + a$
$\alpha^6$	$a \, Z^3 + a^2 \, Z + a^2$	$X^4 + a^2 \, X^3 + a \, X^2 + a^2$
$\alpha^7$	$Z^3 + Z^2 + a^2$	$X^4 + X^3 + a \, X^2 + X + a$
$\alpha^8$	$Z^2 + 1$	$X^4 + X + 1$
$\alpha^9$	$Z^3 + Z^2 + a$	$X^4 + X^3 + a^2 \, X^2 + X + a^2$
$\alpha^{10}$	$a^2 \, Z^3 + a \, Z + a$	$X^4 + a \, X^3 + a^2 \, X^2 + a$
$\alpha^{11}$	$a^2 \, Z^3 + Z + 1$	$X^4 + X^3 + a \, X^2 + a^2$
$\alpha^{12}$	$a \, Z^3 + a \, Z^2 + 1$	$X^4 + a \, X^3 + a \, X^2 + a \, X + a$
$\alpha^{13}$	$a \, Z^2 + a$	$X^4 + a \, X + 1$
$\alpha^{14}$	$a \, Z^3 + a \, Z^2 + a^2$	$X^4 + a \, X^3 + a^2 \, X^2 + a \, X + a^2$

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

Article Metrics

HTML views(1984) PDF downloads(383) Cited by(0)

[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.
[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.
[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.
[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.
[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.
[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.
[7]	S. T. Dougherty, T. A. Gulliver and H. Masaaki, Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.
[8]	M. Giesbrecht, Factoring in skew-polynomial rings over finite fields, J. Symb. Comput., 26 (1998), 463-486. doi: 10.1006/jsco.1998.0224.
[9]	W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511807077.
[10]	N. Jacobson, The Theory of Rings, Amer. Math. Soc., 1943.
[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.
[12]	R. W. K. Odoni, On additive polynomials over a finite field, Proc. Edinburgh Math. Soc., 42 (1999), 1-16. doi: 10.1017/S0013091500019970.
[13]	O. Ore, Theory of Non-Commutative Polynomials, Ann. Math., 34 (1933), 480-508. doi: 10.2307/1968173.
[14]	J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, Cambridge, 2013. doi: 10.1017/CBO9781139856065.
[15]	A. Zhdanov, New self-dual codes of length $72$, preprint, arXiv: 1705.05779.

A first step towards the skew duadic codes

Abstract

References

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

A first step towards the skew duadic codes

Abstract

References

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content