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{{Short description|Algorithm on linear-feedback shift registers}}
{{distinguish|Berlekamp's algorithm}}
[[File:Berlekamp–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>
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:<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 ==
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T(D) T(x) polynomial
-->
polynomial(field K) C(x) = 1; <span class="cm">/* coeffs are c<sub>j</sub> */</span>
polynomial(field K) B(x) = 1;
int L = 0;
int m = 1;
field K b = 1;
int n;
field K d = s<sub>n</sub> + {{math|∑{{su|p=L|b=i=1}} c<sub>i</sub> s<sub>n - i</sub>}} <!--Σi=1Lci⋅sn−i;-->
▲ <span class="w"> </span><span class="cm">/* step 2. calculate discrepancy */</span>
m = m + 1;
▲ <span class="w"> </span><span class="cm">/* step 3. discrepancy is zero; annihilation continues */</span>
polynomial(field K) T(x) = C(x);
▲ <span class="w"> </span><span class="cm">/* temporary copy of C(x) */</span>
C(x) = C(x) - d b<sup>−1</sup> x<sup>m</sup> B(x);
L = n + 1 - L;
B(x) = T(x);
b = d;
m = 1;
} <span class="k">else</span> {
C(x) = C(x) - d b<sup>−1</sup> x<sup>m</sup> B(x);
▲ <span class="w"> </span><span class="cm">/* step 4. */</span>
m = m + 1;
}
}
▲ <span class="w"> </span><span class="p">}</span>
▲ </div>
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/>
/* ... */▼
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 */
}}
==See also==
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==External links==
* {{springer|title=Berlekamp-Massey algorithm|id=p/b120140}}
*
* {{MathWorld|urlname=Berlekamp-MasseyAlgorithm|title=Berlekamp–Massey Algorithm}}
* [https://code.google.com/p/lfsr/ GF(2) implementation in Mathematica]
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