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{{short description|Error correction code}}
In [[coding theory]], the
One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an [[Abstract algebra|algebraic]] method known as [[syndrome decoding]]. This simplifies the design of the decoder for these codes, using small low-power electronic hardware.
BCH codes are used in applications such as satellite communications,<ref>{{cite web|title=Phobos Lander Coding System: Software and Analysis|url=http://ipnpr.jpl.nasa.gov/progress_report/42-94/94V.PDF |
== Definition and illustration ==
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Given a [[prime number]] {{mvar|q}} and [[prime power]] {{math|''q''<sup>''m''</sup>}} with positive integers {{mvar|m}} and {{mvar|d}} such that {{math|''d'' ≤ ''q''<sup>''m''</sup> − 1}}, a primitive narrow-sense BCH code over the [[finite field]] (or Galois field) {{math|GF(''q'')}} with code length {{math|''n'' {{=}} ''q''<sup>''m''</sup> − 1}} and [[Block code#The distance d|distance]] at least {{mvar|d}} is constructed by the following method.
Let {{mvar|α}} be a [[
For any positive integer {{mvar|i}}, let {{math|''m''<sub>''i''</sub>(''x'')}} be the [[minimal polynomial (field theory)|minimal polynomial]] with coefficients in {{math|GF(''q'')}} of {{math|α<sup>''i''</sup>}}.
The [[generator polynomial]] of the BCH code is defined as the [[least common multiple]] {{math|''g''(''x'') {{=}} lcm(''m''<sub>1</sub>(''x''),…,''m''<sub>''d'' − 1</sub>(''x''))}}.
It can be seen that {{math|''g''(''x'')}} is a polynomial with coefficients in {{math|GF(''q'')}} and divides {{math|''x''<sup>''n''</sup> − 1}}.
Therefore, the [[polynomial code]] defined by {{math|''g''(''x'')}} is a cyclic code.
==== Example ====
Let {{math|''q'' {{=}} 2}} and {{math|''m'' {{=}} 4}} (therefore {{math|''n'' {{=}} 15}}). We will consider different values of {{mvar|d}}
:<math>m_i\left(\alpha^i\right) \bmod \left(
The minimal polynomials
:<math>\begin{align}
m_1(x) &= m_2(x) = m_4(x) = m_8(x) = x^4 + x + 1, \\
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\end{align}</math>
The BCH code with <math>d = 2, 3</math> has the generator polynomial
:<math>g(x) = {\rm lcm}(m_1(x), m_2(x)) = m_1(x) = x^4 + x + 1.\,</math>
It has minimal [[Hamming distance]] at least 3 and corrects up to one error. Since the generator polynomial is of degree 4, this code has 11 data bits and 4 checksum bits. It is also denoted as: '''(15, 11) BCH''' code.
The BCH code with <math>d=4,5</math> has the generator polynomial
:<math>\begin{align}
g(x) &= {\rm lcm}(m_1(x),m_2(x),m_3(x),m_4(x)) = m_1(x) m_3(x) \\
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\end{align}</math>
It has minimal Hamming distance at least 5 and corrects up to two errors. Since the generator polynomial is of degree 8, this code has 7 data bits and 8 checksum bits. It is also denoted as: '''(15, 7) BCH''' code.
The BCH code with <math>d=6,7</math> has the generator polynomial
:<math>\begin{align}
g(x) &= {\rm lcm}(m_1(x),m_2(x),m_3(x),m_4(x),m_5(x),m_6(x)) = m_1(x) m_3(x) m_5(x) \\
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\end{align}</math>
It has minimal Hamming distance at least 7 and corrects up to three errors. Since the generator polynomial is of degree 10, this code has 5 data bits and 10 checksum bits. It is also denoted as: '''(15, 5) BCH''' code. (This particular generator polynomial has a real-world application, in the "format
The BCH code with <math>d=8</math> and higher has the generator polynomial
:<math>\begin{align}
g(x) &= {\rm lcm}(m_1(x),m_2(x),...,m_{14}(x)) = m_1(x) m_3(x) m_5(x) m_7(x)\\
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\end{align}</math>
This code has minimal Hamming distance 15 and corrects 7 errors. It has 1 data bit and 14 checksum bits. It is also denoted as: '''(15, 1) BCH''' code. In fact, this code has only two codewords: 000000000000000 and 111111111111111 (a trivial [[repetition code]]).
=== General BCH codes ===
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The generator polynomial <math>g(x)</math> of a BCH code has coefficients from <math>\mathrm{GF}(q).</math>
In general, a cyclic code over <math>\mathrm{GF}(q^p)</math> with <math>g(x)</math> as the generator polynomial is called a BCH code over <math>\mathrm{GF}(q^p).</math>
The BCH code over <math>\mathrm{GF}(q^m)</math> and generator polynomial <math>g(x)</math> with successive powers of <math>\alpha</math> as roots is one type of [[Reed–Solomon code]] where the decoder (syndromes) alphabet is the same as the channel (data and generator polynomial) alphabet, all elements of <math>\mathrm{GF}(q^m)</math> .<ref>{{Harvnb|Gill|n.d.|p=3}}</ref> The other type of Reed Solomon code is an [[
== Properties ==
The generator polynomial of a BCH code has degree at most <math>(d-1)m</math>. Moreover, if <math>q=2</math> and <math>c=1</math>, the generator polynomial has degree at most <math>dm/2</math>.
{{Collapse top|title=Proof}}
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=== Non-systematic encoding: The message as a factor ===
The most straightforward way to find a polynomial that is a multiple of the generator is to compute the product of some arbitrary polynomial and the generator. In this case, the arbitrary polynomial can be chosen using the symbols of the message as coefficients.
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:<math>\begin{align}
s(x) &= p(x)g(x)\\
&= \left(x^{20}+x^{18}+x^{17}+x^{15}+x^{14}+x^{13}+x^{11}+x^{10}+x^9+x^8+x^6+x^5+x^4+x^3+x^2+1\right)\left(x^{10}+x^9+x^8+x^6+x^5+x^3+1\right)\\
&= x^{30}+x^{29}+x^{26}+x^{25}+x^{24}+x^{22}+x^{19}+x^{17}+x^{16}+x^{15}+x^{14}+x^{12}+x^{10}+x^9+x^8+x^6+x^5+x^4+x^2+1
\end{align}</math>
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=== Systematic encoding: The message as a prefix ===
A systematic code is one in which the message appears verbatim somewhere within the codeword. Therefore, systematic BCH encoding involves first embedding the message polynomial within the codeword polynomial, and then adjusting the coefficients of the remaining (non-message) terms to ensure that <math>s(x)</math> is divisible by <math>g(x)</math>.
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The first step is finding, compatible with computed syndromes and with minimal possible <math>t,</math> locator polynomial:
:<math>\Lambda(x) = \prod_{j=1}^t \left(x\alpha^{i_j} - 1\right)</math>
# [[
# [[Berlekamp–Massey algorithm]]
# [[Reed–Solomon error correction#Euclidean decoder|Sugiyama Euclidean algorithm]]
====Peterson–Gorenstein–Zierler algorithm====
<!-- this confuses t (max number of errors that can be corrected) with ν (actual number of errors) -->
[[Peterson's algorithm]] is the step 2 of the generalized BCH decoding procedure. Peterson's algorithm is used to calculate the error locator polynomial coefficients <math> \lambda_1 , \lambda_2, \dots, \lambda_{v} </math> of a polynomial
: <math> \Lambda(x) = 1 + \lambda_1 x + \lambda_2 x^2 + \cdots + \lambda_v x^v .</math>
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set <math>v \leftarrow v -1</math>
continue from the beginning of Peterson's decoding by making smaller <math>S_{v \times v}</math>
| After you have values of <math>\Lambda</math>, you have
| Stop Peterson procedure.
}}
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Let
:<math>S(x) = s_c + s_{c+1}x + s_{c+2}x^2 + \cdots + s_{c+d-2}x^{d-2}.</math>▼
:<math>v \leqslant d-1, \lambda_0 \neq 0 \qquad \Lambda(x) = \sum_{i=0}^v\
▲:<math>S(x)=s_c+s_{c+1}x+s_{c+2}x^2+\cdots+s_{c+d-2}x^{d-2}.</math>
▲:<math>v\leqslant d-1, \lambda_0\neq 0 \qquad \Lambda(x)=\sum_{i=0}^v\lambda_ix^i=\lambda_0 \prod_{k=0}^{v} (\alpha^{-i_k}x-1).</math>
And the error evaluator polynomial<ref name="Gill-Forney">{{Harvnb|Gill|n.d.|p=47}}</ref>
:<math>\Omega(x) \equiv S(x) \Lambda(x) \bmod{x^{d-1}}</math>
Finally:
:<math>\Lambda'(x) = \sum_{i=1}^v i \cdot \lambda_i x^{i-1},</math>
where
:<math>i \cdot x := \sum_{k=1}^i x.</math>▼
▲:<math>i\cdot x := \sum_{k=1}^i x.</math>
Than if syndromes could be explained by an error word, which could be nonzero only on positions <math>i_k</math>, then error values are
:<math>e_k = -{\alpha^{i_k}\Omega\left(\alpha^{-i_k}\right) \over \alpha^{c\cdot i_k}\Lambda'\left(\alpha^{-i_k}\right)}.</math>
For narrow-sense BCH codes, ''c'' = 1, so the expression simplifies to:
:<math>e_k = -{\Omega\left(\alpha^{-i_k}\right) \over \Lambda'\left(\alpha^{-i_k}\right)}.</math>
==== Explanation of Forney algorithm computation ====
It is based on [[Lagrange polynomial|Lagrange interpolation]] and techniques of [[generating function]]s.
Consider <math>S(x)\Lambda(x),</math> and for the sake of simplicity suppose <math>\lambda_k = 0</math> for <math>k > v,</math> and <math>s_k = 0</math> for <math>k > c + d - 2.</math> Then
:<math>S(x)\Lambda(x) = \sum_{j=0}^{\infty}\sum_{i=0}^j s_{j-i+1}\lambda_i x^j.</math>
:<math>\begin{align}
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Thanks to <math>v\leqslant d-1</math> we have
:<math>\Omega(x) = -\lambda_0\sum_{j=1}^v e_j\alpha^{c i_j} \prod_{\ell\in\{1,\cdots,v\}\setminus\{j\}} \left
Thanks to <math>\Lambda</math> (the Lagrange interpolation trick) the sum degenerates to only one summand for <math>x = \alpha^{-i_k}</math>
:<math>\Omega \left
To get <math>e_k</math> we just should get rid of the product. We could compute the product directly from already computed roots <math>\alpha^{-i_j}</math> of <math>\Lambda,</math> but we could use simpler form.
As [[formal derivative]]
:<math>\Lambda'(x) = \lambda_0\sum_{j=1}^v \alpha^{i_j}\prod_{\ell\in\{1,\cdots,v\}\setminus\{j\}} \left
we get again only one summand in
:<math>\Lambda'\left(\alpha^{-i_k}\right) = \lambda_0\alpha^{i_k}\prod_{\ell\in\{1,\cdots,v\}\setminus\{k\}} \left
So finally
:<math>e_k = -\frac{\alpha^{i_k}\Omega \left
This formula is advantageous when one computes the formal derivative of <math>\Lambda</math> form
:<math>\Lambda(x) = \sum_{i=1}^v \
yielding:
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An alternate process of finding both the polynomial Λ and the error locator polynomial is based on Yasuo Sugiyama's adaptation of the [[Extended Euclidean algorithm]].<ref>Yasuo Sugiyama, Masao Kasahara, Shigeichi Hirasawa, and Toshihiko Namekawa. A method for solving key equation for decoding Goppa codes. Information and Control, 27:87–99, 1975.</ref> Correction of unreadable characters could be incorporated to the algorithm easily as well.
Let <math>k_1, ..., k_k</math> be positions of unreadable characters. One creates polynomial localising these positions <math>\Gamma(x) = \prod_{i=1}^k\left(x\alpha^{k_i} - 1\right).</math>
Set values on unreadable positions to 0 and compute the syndromes.
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If <math>\Lambda(x)</math> denotes the polynomial eliminating the influence of these coordinates, we obtain
:<math>S(x)\Gamma(x)\Lambda(x) \stackrel{\{k+v, \cdots, d-2\}}{=} 0.</math>
In Euclidean algorithm, we try to correct at most <math>\tfrac{1}{2}(d-1-k)</math> errors (on readable positions), because with bigger error count there could be more codewords in the same distance from the received word. Therefore, for <math>\Lambda(x)</math> we are looking for, the equation must hold for coefficients near powers starting from
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:<math>R(x) = C(x) + x^{13} + x^5 = x^{14} + x^{11} + x^{10} + x^9 + x^5 + x^4 + x^2</math>
In order to correct the errors, first calculate the syndromes. Taking <math>\alpha = 0010,</math> we have <math>s_1 = R(\alpha^1) = 1011,</math> <math>s_2 = 1001,</math> <math>s_3 = 1011,</math> <math>s_4 = 1101,</math> <math>s_5 = 0001,</math> and <math>s_6 = 1001.</math>
Next, apply the Peterson procedure by row-reducing the following [[augmented matrix]].
:<math>\left [ S_{3 \times 3} | C_{3 \times 1} \right ] =
\begin{bmatrix}s_1&s_2&s_3&s_4\\
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==== Decoding with unreadable characters ====
Suppose the same scenario, but the received word has two unreadable characters [ 1 {{color|red|0}} 0 ? 1 1 ? 0 0 {{color|red|1}} 1 0 1 0 0 ]. We replace the unreadable characters by zeros while creating the
Let us make syndrome polynomial
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:<math>\begin{align}
&\begin{pmatrix}S(x)\Gamma(x)\\ x^6\end{pmatrix}
={} &\begin{pmatrix}\alpha^{-7} +\alpha^{4}x+ \alpha^{-1}x^2+ \alpha^{6}x^3+ \alpha^{-1}x^4+ \alpha^{5}x^5 +\alpha^{7}x^6+ \alpha^{-3}x^7 \\ x^6\end{pmatrix} \\ [6pt] \begin{pmatrix}x^6\\ \alpha^{-7} +\alpha^{4}x +\alpha^{-1}x^2 +\alpha^{6}x^3 +\alpha^{-1}x^4 +\alpha^{5}x^5 +2\alpha^{7}x^6 +2\alpha^{-3}x^7\end{pmatrix} \\ [6pt] \begin{pmatrix}\alpha^4 + \alpha^{-5}x & 1\\ 1 & 0\end{pmatrix} &\qquad
\begin{pmatrix} \begin{pmatrix}\alpha^{-5}+ \alpha^{-4}x & 1\\ 1 & 0 \end{pmatrix} &\qquad
\begin{pmatrix}
\end{align}</math>
We have reached polynomial of degree at most 3, and as
:<math>\begin{pmatrix}-\left(\alpha^4+ \alpha^{-5}x\right) & \alpha^{-3}+ \alpha^{5}x+ \alpha^{7}x^2\\ \alpha^{3}+ \alpha^{-5}x+ \alpha^{6}x^2 & -\left(\alpha^{7}x+ \alpha^{5}x^2+ \alpha^{3}x^3\right)\end{pmatrix} \begin{pmatrix}
we get
:<math>
\begin{pmatrix}S(x)\Gamma(x)\\ x^6\end{pmatrix} = \begin{pmatrix}
Therefore,
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:<math>S(x)\Gamma(x)\left(\alpha^{3} + \alpha^{-5}x + \alpha^{6}x^2\right) - \left(\alpha^{7}x + \alpha^{5}x^2 + \alpha^{3}x^3\right)x^6 = \alpha^{-4} + \alpha^{4}x + \alpha^{2}x^2 + \alpha^{-5}x^3.</math>
Let <math>\Lambda(x) = \alpha^{3}+ \alpha^{-5}x+ \alpha^{6}x^2.</math> Don't worry that <math>\lambda_0\neq 1.</math> Find by brute force a root of <math>\Lambda.</math> The roots are <math>\alpha^2,</math> and <math>\alpha^{10}</math> (after finding for example <math>\alpha^2</math> we can divide <math>\Lambda</math> by corresponding monom <math>\left(x - \alpha^2\right)</math> and the root of resulting monom could be found easily).
Let
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:<math>\begin{align}
\Xi(x) &= \Gamma(x)\Lambda(x) = \alpha^3 + \alpha^4x^2 + \alpha^2x^3 + \alpha^{-5}x^4 \\
\Omega(x) &= S(x)\Xi(x) \equiv \alpha^{-4} + \alpha^4x + \alpha^2x^2 + \alpha^{-5}x^3 \bmod{x^6}
\end{align}</math>
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:<math>\begin{align}
e_1 &=-\frac{\Omega(\alpha^4)}{\Xi'(\alpha^{4})} = \frac{\alpha^{-4}+\alpha^{-7}+\alpha^{-5}+\alpha^{7}}{\alpha^{-5}} =\frac{\alpha^{-5}}{\alpha^{-5}}=1 \\
e_2 &=-\frac{\Omega(\alpha^7)}{\Xi'(\alpha^{7})} = \frac{\alpha^{-4}+\alpha^{-4}+\alpha^{1}+\alpha^{1}}{\alpha^{1}}=0 \\
e_3 &=-\frac{\Omega(\alpha^{10})}{\Xi'(\alpha^{10})} = \frac{\alpha^{-4}+\alpha^{-1}+\alpha^{7}+\alpha^{-5}}{\alpha^{7}}=\frac{\alpha^{7}}{\alpha^{7}}=1 \\
e_4 &=-\frac{\Omega(\alpha^{2})}{\Xi'(\alpha^{2})} = \frac{\alpha^{-4}+\alpha^{6}+\alpha^{6}+\alpha^{1}}{\alpha^{6}}=\frac{\alpha^{6}}{\alpha^{6}}=1
\end{align}</math>
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Let us show the algorithm behaviour for the case with small number of errors. Let the received word is [ 1 {{color|red|0}} 0 ? 1 1 ? 0 0 0 1 0 1 0 0 ].
Again, replace the unreadable characters by zeros while creating the
Compute the syndromes <math>s_1 = \alpha^{4}, s_2 = \alpha^{-7}, s_3 = \alpha^{1}, s_4 = \alpha^{1}, s_5 = \alpha^{0},</math> and <math>s_6 = \alpha^{2}.</math>
Create syndrome polynomial
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|author-link= Alexis Hocquenghem
|title=Codes correcteurs d'erreurs
|language=
|journal= Chiffres
|___location= Paris
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|pages= 147–156
|date= September 1959
}}
* {{Citation
|
|
|author-link= R. C. Bose
|first2= D. K.
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|date= March 1960
|issn= 0890-5401
|doi=10.1016/s0019-9958(60)90287-4|url= http://repository.lib.ncsu.edu/bitstream/1840.4/2137/1/ISMS_1959_240.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://repository.lib.ncsu.edu/bitstream/1840.4/2137/1/ISMS_1959_240.pdf |archive-date=2022-10-09 |url-status=live
}}
===Secondary sources===
* {{Citation|last=Gill |first=John |title=EE387 Notes #7, Handout #28 |date=n.d. |
* {{Citation
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|last2= Peterson
|first2= W. Wesley
|
|last3= Zierler
|first3 = Neal
|
|title= Two-Error Correcting Bose-Chaudhuri Codes are Quasi-Perfect
|journal= Information and Control
Line 653 ⟶ 652:
|pages= 291–294
|year= 1960
|doi= 10.1016/s0019-9958(60)90877-9
}}
* {{Citation
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|first2= Günter
|last2= Pilz
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|publisher= John Wiley
|year= 1999
}}
* {{Citation
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|first2= Xuemin
|last2= Chen
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|year= 1999
|isbn= 0-7923-8528-4
==Further reading==
* {{Citation |last1=Blahut |first1=Richard E. |author-link1=Richard Blahut |title=Algebraic Codes for Data Transmission |edition=2nd |publisher=[[Cambridge University Press]] |year=2003 |isbn=0-521-55374-1}}
* {{Citation
|
|
|first2= W. K.
|last2= Nicholson
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|publisher= John Wiley
|year= 2004
}}
* {{Citation
|
|
|first2= D.
|last2= Costello
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|___location= Englewood Cliffs, NJ
|year= 2004
}}
* {{Citation
|
|
|first2= N. J. A.
|last2= Sloane
|
|title= The Theory of Error-Correcting Codes
|___location= New York, NY
|publisher= North-Holland Publishing Company
|year= 1977
}}
* {{Citation
|first = Atri
Line 722 ⟶ 716:
|publisher = University at Buffalo
|url = http://www.cse.buffalo.edu/~atri/courses/coding-theory/
|archive-url = https://web.archive.org/web/
▲ |archive-url = https://web.archive.org/web/20100702120650/http://www.cse.buffalo.edu/~atri/courses/coding-theory/
▲ |archive-date = 2010-07-02
|url-status = dead
}}
| |||