Berlekamp–Massey algorithm: Difference between revisions

Revision as of 03:42, 22 November 2024

Latest revision as of 06:47, 3 May 2025

(6 intermediate revisions by 5 users not shown)

Line 1:

{{Short description|Algorithm on linear-feedback shift registers}}

{{distinguish|Berlekamp's algorithm}}

[[File:Berlekamp–Massey_algorithmBerlekamp–Massey algorithm.png | thumb | right | Berlekamp–Massey algorithm]]

The '''Berlekamp–Massey algorithm''' is an [[algorithm]] that will find the shortest [[linear-feedback shift register]] (LFSR) for a given binary output sequence. The algorithm will also find the [[Minimal polynomial (field theory)|minimal polynomial]] of a linearly [[Recurrence relation|recurrent sequence]] in an arbitrary [[field (mathematics)|field]]. The field requirement means that the Berlekamp–Massey algorithm requires all non-zero elements to have a multiplicative inverse.<ref>{{Harvnb|Reeds|Sloane|1985|p=2}}</ref> Reeds and Sloane offer an extension to handle a [[ring (mathematics)|ring]].<ref>{{Citation |last1=Reeds |first1=J. A. |last2=Sloane |first2=N. J. A. |author-link2=N. J. A. Sloane |journal=SIAM Journal on Computing |volume=14 |issue=3 |pages=505–513 |year=1985 |title=Shift-Register Synthesis (Modulo n) |url=http://neilsloane.com/doc/Me111.pdf |doi=10.1137/0214038 |citeseerx=10.1.1.48.4652 }}</ref>

Line 90:

:<math> d = d - (d/b)b = d - d = 0.</math>

The algorithm also needs to increase ''L'' (number of errors) as needed. If ''L'' equals the actual number of errors, then during the iteration process, the discrepancies will become zero before ''n'' becomes greater than or equal to 2''L''. Otherwise ''L'' is updated and the algorithm will update ''B''(''x''), ''b'', increase ''L'', and reset ''m'' = 1. The formula ''L'' = (''n'' + 1 − ''L'') limits ''L'' to the number of available syndromes used to calculate discrepancies, and also handles the case where ''L'' increases by more than 1.

== Pseudocode ==

Line 104:

T(D) T(x) polynomial

-->

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr">

polynomial(field ''K'') s(x) = ... <span class="cm">/* coeffs are s_js_j; output sequence as N-1 degree polynomial) */</span>

<syntaxhighlight lang="c">

<span class="cm">/* connection polynomial */</span>

polynomial(field K) s(x) = ... /* coeffs are s_j; output sequence as N-1 degree polynomial) */

polynomial(field K) C(x) = 1; <span class="cm">/* coeffs are c_j */</span>

/* connection polynomial */

polynomial(field K) CB(x) = 1; /* coeffs are c_j */

int mL = m + 10;

polynomial(field K) B(x) = 1;

int Lm = 01;

int m field K b = 1;

field K b = 1int n;

<span class="cm">/* steps 2. and 6. */</span>

int n;

<span class="k">for</span> (n = 0; n < N; n++) {

<span class="cm">/* step 2. calculate discrepancy */</span>

/* steps 2. and 6. */

field K d = s_n + {{math|&sum;{{su|p=L|b=i=1}} c_i s_{n - i}}} <!--Σi=1Lci⋅sn−i;-->

for (n = 0; n < N; n++) {

/* step 2. calculate discrepancy */

<span class="k">if</span> (d == 0) {

field K d = s_n + ∑i=1Lci⋅sn−i;

<span class="cm">/* step 3. discrepancy is zero; annihilation continues */</span>

if (d m == 0)m + {1;

} <span class="k">else</span> <span class="k">if</span> (2 * L <= n) {

/* step 3. discrepancy is zero; annihilation continues */

m <span class="cm">/* mstep +5. 1;*/</span>

} else if (2 * L <span class="cm">/* ntemporary copy of C(x) {*/</span>

/* step 5. */ polynomial(field K) T(x) = C(x);

/* temporary copy of C(x) */

polynomial(field K) T C(x) = C(x) - d b⁻¹ x^m B(x);

L = n + 1 - L;

C(x) = C B(x) -= d b−1xm BT(x);

L = n + 1b -= Ld;

B(x) m = T(x)1;

b} <span class="k">else</span> d;{

m <span class="cm">/* step 4. 1;*/</span>

C(x) = C(x) - d b⁻¹ x^m B(x);

} else {

/* step 4. */ m = m + 1;

C(x) = C(x) - d b−1xm B(x);}

m = m + 1;

}

<span class="k">return</span> L;

}

</div>

return L;

</syntaxhighlight>

In the case of binary GF(2) BCH code, the discrepancy d will be zero on all odd steps, so a check can be added to avoid calculating it.

{{sxhl|2=c|1=<nowiki/>

}/* else... {*/

<syntaxhighlight lang="c">

for (n = 0; n < N; n++) {

/* ... */

/* if odd step number, discrepancy == 0, no need to calculate it */

for (n = 0; n < N; n++) {

if ((n&1) != 0) {

/* if odd step number, discrepancy == 0, no need to calculate it */

if ((n&1) ! m = 0)m {+ 1;

m = m + 1continue;

continue;}

}/* ... */

}}

/* ... */

</syntaxhighlight>

==See also==

Line 165 ⟶ 162:

==External links==

* {{springer|title=Berlekamp-Massey algorithm|id=p/b120140}}

* [https://web.archive.org/web/20120716181541/http://planetmath.org/encyclopedia/{{PlanetMath|BerlekampMasseyAlgorithm.html |Berlekamp–Massey algorithm] at [[PlanetMath]].}}

* {{MathWorld|urlname=Berlekamp-MasseyAlgorithm|title=Berlekamp–Massey Algorithm}}

* [https://code.google.com/p/lfsr/ GF(2) implementation in Mathematica]