Constructing generator polynomial for a BCH code
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How can I construct a generator polynomial for a BCH code $(7,3)$ code
over $GF(2^3)$ with designed distance $delta =5$.
Observe that $$x^7-1 = (x+1)(x^3+x+1)(x^3+x^2+1)=m_0(x)m_1(x)m_2(x)$$ where $m_i(x)$ are the minimal polynomials for $alpha^i$ where $alpha$ is the primitive element of $$GF(2^3)= GF(2)/(y^3+y+1).$$ Also, $alpha$ is the $7$-th root of unity and $m_1(x)=m_2(x)=m_4(x)$ and $m_3=m_5=m_6(x)$.
Note that the cyclotomic sets are $$C_0={0}, quad C_1= {1,2,4}, C_2= {3,5,6}.$$ Since the designed distance is $5$, I cant pick $delta-1=4$ distinct consecutive elements from $$alpha^a, alpha^{a+1}, cdots, alpha^{a+delta-2}$$ for any value of $a$ without having an element in the three cosets and that will make the degree of my polynomial to be $6$ or $7$ and I also want the weight of the generator to be at least $5$. The most natural choice would be $$1+x+x^2+x^3+x^4$$ but this can not work because it does not divide $x^7-1$. Also, it is not the lcm of any of the minimal polynomial. Please, How can i construct this generator polynomial. Any help will be appreciated.
abstract-algebra coding-theory
asked Nov 26 '18 at 7:33
Jaynot
487515
add a comment |
How can I construct a generator polynomial for a BCH code $(7,3)$ code
over $GF(2^3)$ with designed distance $delta =5$.
Observe that $$x^7-1 = (x+1)(x^3+x+1)(x^3+x^2+1)=m_0(x)m_1(x)m_2(x)$$ where $m_i(x)$ are the minimal polynomials for $alpha^i$ where $alpha$ is the primitive element of $$GF(2^3)= GF(2)/(y^3+y+1).$$ Also, $alpha$ is the $7$-th root of unity and $m_1(x)=m_2(x)=m_4(x)$ and $m_3=m_5=m_6(x)$.
Note that the cyclotomic sets are $$C_0={0}, quad C_1= {1,2,4}, C_2= {3,5,6}.$$ Since the designed distance is $5$, I cant pick $delta-1=4$ distinct consecutive elements from $$alpha^a, alpha^{a+1}, cdots, alpha^{a+delta-2}$$ for any value of $a$ without having an element in the three cosets and that will make the degree of my polynomial to be $6$ or $7$ and I also want the weight of the generator to be at least $5$. The most natural choice would be $$1+x+x^2+x^3+x^4$$ but this can not work because it does not divide $x^7-1$. Also, it is not the lcm of any of the minimal polynomial. Please, How can i construct this generator polynomial. Any help will be appreciated.
abstract-algebra coding-theory
asked Nov 26 '18 at 7:33
Jaynot
487515
2
You need to work over the field $GF(8)$, not $GF(2)$.
– Wuestenfux
Nov 26 '18 at 7:51
@Wuestenfux Thank you for pointing that out. I totally forgot and I figured it out. In fact, the cyclotomic sets are singletons and the generator polynomial is trivial. Thanks once again :)
– Jaynot
Nov 26 '18 at 8:32
add a comment |
How can I construct a generator polynomial for a BCH code $(7,3)$ code
over $GF(2^3)$ with designed distance $delta =5$.
Observe that $$x^7-1 = (x+1)(x^3+x+1)(x^3+x^2+1)=m_0(x)m_1(x)m_2(x)$$ where $m_i(x)$ are the minimal polynomials for $alpha^i$ where $alpha$ is the primitive element of $$GF(2^3)= GF(2)/(y^3+y+1).$$ Also, $alpha$ is the $7$-th root of unity and $m_1(x)=m_2(x)=m_4(x)$ and $m_3=m_5=m_6(x)$.
Note that the cyclotomic sets are $$C_0={0}, quad C_1= {1,2,4}, C_2= {3,5,6}.$$ Since the designed distance is $5$, I cant pick $delta-1=4$ distinct consecutive elements from $$alpha^a, alpha^{a+1}, cdots, alpha^{a+delta-2}$$ for any value of $a$ without having an element in the three cosets and that will make the degree of my polynomial to be $6$ or $7$ and I also want the weight of the generator to be at least $5$. The most natural choice would be $$1+x+x^2+x^3+x^4$$ but this can not work because it does not divide $x^7-1$. Also, it is not the lcm of any of the minimal polynomial. Please, How can i construct this generator polynomial. Any help will be appreciated.
abstract-algebra coding-theory
asked Nov 26 '18 at 7:33
Jaynot
487515
How can I construct a generator polynomial for a BCH code $(7,3)$ code
over $GF(2^3)$ with designed distance $delta =5$.
Observe that $$x^7-1 = (x+1)(x^3+x+1)(x^3+x^2+1)=m_0(x)m_1(x)m_2(x)$$ where $m_i(x)$ are the minimal polynomials for $alpha^i$ where $alpha$ is the primitive element of $$GF(2^3)= GF(2)/(y^3+y+1).$$ Also, $alpha$ is the $7$-th root of unity and $m_1(x)=m_2(x)=m_4(x)$ and $m_3=m_5=m_6(x)$.
Note that the cyclotomic sets are $$C_0={0}, quad C_1= {1,2,4}, C_2= {3,5,6}.$$ Since the designed distance is $5$, I cant pick $delta-1=4$ distinct consecutive elements from $$alpha^a, alpha^{a+1}, cdots, alpha^{a+delta-2}$$ for any value of $a$ without having an element in the three cosets and that will make the degree of my polynomial to be $6$ or $7$ and I also want the weight of the generator to be at least $5$. The most natural choice would be $$1+x+x^2+x^3+x^4$$ but this can not work because it does not divide $x^7-1$. Also, it is not the lcm of any of the minimal polynomial. Please, How can i construct this generator polynomial. Any help will be appreciated.
abstract-algebra coding-theory
abstract-algebra coding-theory
asked Nov 26 '18 at 7:33
Jaynot
487515
asked Nov 26 '18 at 7:33
Jaynot
487515
asked Nov 26 '18 at 7:33
Jaynot
487515
asked Nov 26 '18 at 7:33
Jaynot
487515
asked Nov 26 '18 at 7:33
Jaynot
487515
487515
2
You need to work over the field $GF(8)$, not $GF(2)$.
– Wuestenfux
Nov 26 '18 at 7:51
@Wuestenfux Thank you for pointing that out. I totally forgot and I figured it out. In fact, the cyclotomic sets are singletons and the generator polynomial is trivial. Thanks once again :)
– Jaynot
Nov 26 '18 at 8:32
add a comment |
2
You need to work over the field $GF(8)$, not $GF(2)$.
– Wuestenfux
Nov 26 '18 at 7:51
@Wuestenfux Thank you for pointing that out. I totally forgot and I figured it out. In fact, the cyclotomic sets are singletons and the generator polynomial is trivial. Thanks once again :)
– Jaynot
Nov 26 '18 at 8:32
2
2
You need to work over the field $GF(8)$, not $GF(2)$.
– Wuestenfux
Nov 26 '18 at 7:51
You need to work over the field $GF(8)$, not $GF(2)$.
– Wuestenfux
Nov 26 '18 at 7:51
@Wuestenfux Thank you for pointing that out. I totally forgot and I figured it out. In fact, the cyclotomic sets are singletons and the generator polynomial is trivial. Thanks once again :)
– Jaynot
Nov 26 '18 at 8:32
@Wuestenfux Thank you for pointing that out. I totally forgot and I figured it out. In fact, the cyclotomic sets are singletons and the generator polynomial is trivial. Thanks once again :)
– Jaynot
Nov 26 '18 at 8:32
add a comment |
1 Answer
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The first thing you have to do is to create a Field with 8 elements and find a primite element of it. For this, it's sufficient to consider $mathbb{F}_8 := mathbb{F}_2/(x^3+x+1)$. Denote $alpha := [x]$, then $alpha^3=alpha+1$. Finding a primitive element of the field means find a generator of the multiplicative group $(mathbb{F}_8)^* := mathbb{F}_8$. Note that $vert (mathbb{F}_8)^* vert = 7$ and since every element $ 1 neq beta in (mathbb{F}_8)^*$ has a period that divide 7, each non trivial element of $(mathbb{F}_8)^*$ is primitive. In conclusion we can take $alpha$ as a primitive element of the field.
Since your code is a Reed Solomon code, the cyclotomic sets are made up by just one element, and you can consider the Defining set ${ 1,2,3,4 }$, which is also a complete defining set for a suitable polynomial $$g:=(x-alpha)cdot (x-alpha^2) cdot (x-alpha^3) cdot (x-alpha^4) in mathbb{F}_8[x]$$
Computing $g$ we obtain $g = x^4+(alpha+1)x^3+x^2+alpha x + alpha + 1$
Observe that this polynomial satisfies your request
answered Nov 26 '18 at 13:24
Bilo
1089
add a comment |
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The first thing you have to do is to create a Field with 8 elements and find a primite element of it. For this, it's sufficient to consider $mathbb{F}_8 := mathbb{F}_2/(x^3+x+1)$. Denote $alpha := [x]$, then $alpha^3=alpha+1$. Finding a primitive element of the field means find a generator of the multiplicative group $(mathbb{F}_8)^* := mathbb{F}_8$. Note that $vert (mathbb{F}_8)^* vert = 7$ and since every element $ 1 neq beta in (mathbb{F}_8)^*$ has a period that divide 7, each non trivial element of $(mathbb{F}_8)^*$ is primitive. In conclusion we can take $alpha$ as a primitive element of the field.
Since your code is a Reed Solomon code, the cyclotomic sets are made up by just one element, and you can consider the Defining set ${ 1,2,3,4 }$, which is also a complete defining set for a suitable polynomial $$g:=(x-alpha)cdot (x-alpha^2) cdot (x-alpha^3) cdot (x-alpha^4) in mathbb{F}_8[x]$$
Computing $g$ we obtain $g = x^4+(alpha+1)x^3+x^2+alpha x + alpha + 1$
Observe that this polynomial satisfies your request
answered Nov 26 '18 at 13:24
Bilo
1089
add a comment |
The first thing you have to do is to create a Field with 8 elements and find a primite element of it. For this, it's sufficient to consider $mathbb{F}_8 := mathbb{F}_2/(x^3+x+1)$. Denote $alpha := [x]$, then $alpha^3=alpha+1$. Finding a primitive element of the field means find a generator of the multiplicative group $(mathbb{F}_8)^* := mathbb{F}_8$. Note that $vert (mathbb{F}_8)^* vert = 7$ and since every element $ 1 neq beta in (mathbb{F}_8)^*$ has a period that divide 7, each non trivial element of $(mathbb{F}_8)^*$ is primitive. In conclusion we can take $alpha$ as a primitive element of the field.
Since your code is a Reed Solomon code, the cyclotomic sets are made up by just one element, and you can consider the Defining set ${ 1,2,3,4 }$, which is also a complete defining set for a suitable polynomial $$g:=(x-alpha)cdot (x-alpha^2) cdot (x-alpha^3) cdot (x-alpha^4) in mathbb{F}_8[x]$$
Computing $g$ we obtain $g = x^4+(alpha+1)x^3+x^2+alpha x + alpha + 1$
Observe that this polynomial satisfies your request
answered Nov 26 '18 at 13:24
Bilo
1089
add a comment |
The first thing you have to do is to create a Field with 8 elements and find a primite element of it. For this, it's sufficient to consider $mathbb{F}_8 := mathbb{F}_2/(x^3+x+1)$. Denote $alpha := [x]$, then $alpha^3=alpha+1$. Finding a primitive element of the field means find a generator of the multiplicative group $(mathbb{F}_8)^* := mathbb{F}_8$. Note that $vert (mathbb{F}_8)^* vert = 7$ and since every element $ 1 neq beta in (mathbb{F}_8)^*$ has a period that divide 7, each non trivial element of $(mathbb{F}_8)^*$ is primitive. In conclusion we can take $alpha$ as a primitive element of the field.
Since your code is a Reed Solomon code, the cyclotomic sets are made up by just one element, and you can consider the Defining set ${ 1,2,3,4 }$, which is also a complete defining set for a suitable polynomial $$g:=(x-alpha)cdot (x-alpha^2) cdot (x-alpha^3) cdot (x-alpha^4) in mathbb{F}_8[x]$$
Computing $g$ we obtain $g = x^4+(alpha+1)x^3+x^2+alpha x + alpha + 1$
Observe that this polynomial satisfies your request
answered Nov 26 '18 at 13:24
Bilo
1089
The first thing you have to do is to create a Field with 8 elements and find a primite element of it. For this, it's sufficient to consider $mathbb{F}_8 := mathbb{F}_2/(x^3+x+1)$. Denote $alpha := [x]$, then $alpha^3=alpha+1$. Finding a primitive element of the field means find a generator of the multiplicative group $(mathbb{F}_8)^* := mathbb{F}_8$. Note that $vert (mathbb{F}_8)^* vert = 7$ and since every element $ 1 neq beta in (mathbb{F}_8)^*$ has a period that divide 7, each non trivial element of $(mathbb{F}_8)^*$ is primitive. In conclusion we can take $alpha$ as a primitive element of the field.
Since your code is a Reed Solomon code, the cyclotomic sets are made up by just one element, and you can consider the Defining set ${ 1,2,3,4 }$, which is also a complete defining set for a suitable polynomial $$g:=(x-alpha)cdot (x-alpha^2) cdot (x-alpha^3) cdot (x-alpha^4) in mathbb{F}_8[x]$$
Computing $g$ we obtain $g = x^4+(alpha+1)x^3+x^2+alpha x + alpha + 1$
Observe that this polynomial satisfies your request
answered Nov 26 '18 at 13:24
Bilo
1089
edited Nov 26 '18 at 13:37
answered Nov 26 '18 at 13:24
Bilo
1089
answered Nov 26 '18 at 13:24
Bilo
1089
answered Nov 26 '18 at 13:24
Bilo
1089
1089
add a comment |
add a comment |
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Qqkg,gpS,6pTNvBiSKeH,SYnsryKt4vzE,GC
2
You need to work over the field $GF(8)$, not $GF(2)$.
– Wuestenfux
Nov 26 '18 at 7:51
@Wuestenfux Thank you for pointing that out. I totally forgot and I figured it out. In fact, the cyclotomic sets are singletons and the generator polynomial is trivial. Thanks once again :)
– Jaynot
Nov 26 '18 at 8:32